id int64 | problem_number string | title string | statement string | background string | difficulty_level_id int64 | status string | proposed_by string | proposed_year int64 | category_id int64 | set_id int64 | view_count int64 | favorite_count int64 | created_at timestamp[s] | updated_at timestamp[s] | published bool | category dict | difficulty dict | set dict | duplicates list | status_review unknown | archive_reason string | duplicate_of list | old_problem_number string | notes string | tags list | literature_assessment string | literature_sources list | literature_checked_at timestamp[s] | solution_reference string | source_url string | source string | prize string | references list | clean_statement string | original_statement string | statement_status string | statement_verification string | research_classification string | research_summary string | research_difficulty_suggested string | historical_research_summary string | research_summary_reviewed_at timestamp[s] | source_citation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2,464 | EP-1063 | Erdős Problem #1063 | Let $k\geq 2$ and define $n_k\geq 2k$ to be the least value of $n$ such that $n-i$ divides $\binom{n}{k}$ for all but one $0\leq i<k$. Estimate $n_k$. | Erdős Problems (#1063, https://www.erdosproblems.com/1063). Maintainer status: OPEN. OEIS: A389360. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1063). | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1063",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1063 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,465 | EP-1065 | Erdős Problem #1065 | Are there infinitely many primes $p$ such that $p=2^kq+1$ for some prime $q$ and $k\geq 0$? Or $p=2^k3^lq+1$? | Erdős Problems (#1065, https://www.erdosproblems.com/1065). Maintainer status: OPEN. OEIS: A074781, A339465. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1065). | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1065",
"checked_at": "2026-10-16",
"kind": "open",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1065 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,466 | EP-1066 | Erdős Problem #1066 | Let $G$ be a graph given by $n$ points in $\mathbb{R}^2$, where any two distinct points are at least distance $1$ apart, and we draw an edge between two points if they are distance $1$ apart.
Let $g(n)$ be maximal such that any such graph always has an independent set on at least $g(n)$ vertices. Estimate $g(n)$, or p... | Erdős Problems (#1066, https://www.erdosproblems.com/1066). Maintainer status: OPEN. | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1066",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"planar-graphs"
] | null | null | null | null | https://www.erdosproblems.com/1066 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,467 | EP-1068 | Erdős Problem #1068 | Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely vertex-connected? | Erdős Problems (#1068, https://www.erdosproblems.com/1068). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1068). | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1068",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"set-theory",
"chromatic-number"
] | null | null | null | null | https://www.erdosproblems.com/1068 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,468 | EP-1070 | Erdős Problem #1070 | Let $f(n)$ be maximal such that, given any $n$ points in $\mathbb{R}^2$, there exist $f(n)$ points such that no two are distance $1$ apart. Estimate $f(n)$. In particular, is it true that $f(n)\geq n/4$? | Erdős Problems (#1070, https://www.erdosproblems.com/1070). Maintainer status: OPEN. | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1070",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"geometry"
] | null | null | null | null | https://www.erdosproblems.com/1070 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,469 | EP-1071 | Erdős Problem #1071 | Is there a finite set of unit line segments (rotated and translated copies of $(0,1)$) in the unit square, no two of which intersect, which are maximal with respect to this property?
Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite? | Erdős Problems (#1071, https://www.erdosproblems.com/1071). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1071). | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1071",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"geometry"
] | The maintained Erdős Problems source lists problem 1071 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1071",
"label": "Erdős Problems #1071: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://hugg... | 2026-09-27T00:00:00 | Erdős Problems #1071: proved (Lean): https://www.erdosproblems.com/1071
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | https://www.erdosproblems.com/1071 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,470 | EP-1072 | Erdős Problem #1072 | For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\equiv 0\pmod{p}$.
Is it true that there are infinitely many $p$ for which $f(p)=p-1$?
Is it true that $f(p)/p\to 0$ for almost all $p$? | Erdős Problems (#1072, https://www.erdosproblems.com/1072). Maintainer status: OPEN. OEIS: A073944, A072937, A154554. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1072). | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1072",
"checked_at": "2026-10-16",
"kind": "open",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1072 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,471 | EP-1073 | Erdős Problem #1073 | Let $A(x)$ count the number of composite $u<x$ such that $n!+1\equiv 0\pmod{u}$ for some $n$. Is it true that $A(x)\leq x^{o(1)}$? | Erdős Problems (#1073, https://www.erdosproblems.com/1073). Maintainer status: OPEN. OEIS: A256519. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1073). | 1 | open | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1073",
"checked_at": "2026-10-16",
"kind": "open",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1073 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,472 | EP-1074 | Erdős Problem #1074 | Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$. Does\[\lim \frac{\lvert S\cap [1,x]\rvert}{x}\]exist? What is it?
Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv ... | Erdős Problems (#1074, https://www.erdosproblems.com/1074). Maintainer status: OPEN. OEIS: A063980, A064164. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1074). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1074",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1074 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,473 | EP-1075 | Erdős Problem #1075 | Let $r\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\epsilon>0$, if $n$ is sufficiently large, the following holds.
Any $r$-uniform hypergraph on $n$ vertices with at least $(1+\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\to \infty$ as $n\to \inf... | Erdős Problems (#1075, https://www.erdosproblems.com/1075). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1075). | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1075",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"hypergraphs"
] | null | null | null | null | https://www.erdosproblems.com/1075 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,474 | EP-1083 | Erdős Problem #1083 | Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances. Estimate $f_d(n)$ - in particular, is it true that\[f_d(n)=n^{\frac{2}{d}-o(1)}?\] | Erdős Problems (#1083, https://www.erdosproblems.com/1083). Maintainer status: OPEN. OEIS: A186704. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1083). | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1083",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | [Discussion #4 literature review, 2026-10-08] `10.1007/s00493-008-2357-3` → `10.1007/s00493-008-2099-1`. | [
"erdos-problems",
"geometry",
"distances"
] | null | null | null | null | https://www.erdosproblems.com/1083 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,475 | EP-1084 | Erdős Problem #1084 | Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\mathbb{R}^d$, all of distance at least $1$ apart, there are at most $f_d(n)$ many pairs of points which are distance $1$ apart. Estimate $f_d(n)$. | Erdős Problems (#1084, https://www.erdosproblems.com/1084). Maintainer status: OPEN. contact number problem OEIS: A045945. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1084). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1084",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | contact number problem | [
"erdos-problems",
"geometry",
"distances"
] | null | null | null | null | https://www.erdosproblems.com/1084 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,476 | EP-1085 | Erdős Problem #1085 | Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $f_d(n)$. | Erdős Problems (#1085, https://www.erdosproblems.com/1085). Maintainer status: OPEN. OEIS: A186705. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1085). | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1085",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | [Discussion #4 literature review, 2026-10-08] `10.1007/s00454-008-9095-7` → `10.1007/s00454-008-9082-x`. | [
"erdos-problems",
"geometry",
"distances"
] | null | null | null | null | https://www.erdosproblems.com/1085 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,477 | EP-1086 | Erdős Problem #1086 | Let $g(n)$ be minimal such that any set of $n$ points in $\mathbb{R}^2$ contains the vertices of at most $g(n)$ many triangles with the same area. Estimate $g(n)$. | Erdős Problems (#1086, https://www.erdosproblems.com/1086). Maintainer status: OPEN. | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1086",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"geometry",
"distances"
] | null | null | null | null | https://www.erdosproblems.com/1086 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,478 | EP-1087 | Erdős Problem #1087 | Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\leq n^{3+o(1)}$? | Erdős Problems (#1087, https://www.erdosproblems.com/1087). Maintainer status: OPEN. | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1087",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"geometry",
"distances"
] | null | null | null | null | https://www.erdosproblems.com/1087 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,479 | EP-1088 | Erdős Problem #1088 | Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points such that any two determined distances are distinct. Estimate $f_d(n)$. In particular, is it true that, for fixed $n\geq 3$,\[f_d(n)=2^{o(d)}?\] | Erdős Problems (#1088, https://www.erdosproblems.com/1088). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1088). | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1088",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"geometry"
] | null | null | null | null | https://www.erdosproblems.com/1088 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,480 | EP-1089 | Erdős Problem #1089 | Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\mathbb{R}^d$ determines at least $n$ many distinct distances. Estimate $g_d(n)$. In particular, does\[\lim_{d\to \infty}\frac{g_d(n)}{d^{n-1}}\]exist? | Erdős Problems (#1089, https://www.erdosproblems.com/1089). Maintainer status: SOLVED. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1089). | 1 | solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1089",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "solved"
} | null | null | null | null | [
"erdos-problems",
"geometry",
"distances"
] | The maintained Erdős Problems source lists problem 1089 as “solved” on 2026-09-27. No Lean verification is inferred from the existence of a formalized statement.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1089",
"label": "Erdős Problems #1089: solved"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingface... | 2026-09-27T00:00:00 | Erdős Problems #1089: solved: https://www.erdosproblems.com/1089
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | https://www.erdosproblems.com/1089 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,481 | EP-1091 | Erdős Problem #1091 | Let $G$ be a $K_4$-free graph with chromatic number $4$. Must $G$ contain an odd cycle with at least two diagonals?
More generally, is there some $f(r)\to \infty$ such that every graph with chromatic number $4$, in which every subgraph on $\leq r$ vertices has chromatic number $\leq 3$, contains an odd cycle with at l... | Erdős Problems (#1091, https://www.erdosproblems.com/1091). Maintainer status: SOLVED. | 1 | solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1091",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "solved"
} | null | null | null | null | [
"erdos-problems",
"chromatic-number",
"graph-theory"
] | The maintained Erdős Problems source lists problem 1091 as “solved” on 2026-09-27. No Lean verification is inferred from the existence of a formalized statement.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1091",
"label": "Erdős Problems #1091: solved"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingface... | 2026-09-27T00:00:00 | Erdős Problems #1091: solved: https://www.erdosproblems.com/1091
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | https://www.erdosproblems.com/1091 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,482 | EP-1092 | Erdős Problem #1092 | Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $\leq r$ and a graph with $\leq f_r(m)$ edges, then $G$ has chromatic number $\leq r+1$.
Is it true that $f_2(n) \gg n$? More generally, is $f_r(n)\gg_r n$? | Erdős Problems (#1092, https://www.erdosproblems.com/1092). Maintainer status: DISPROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1092). | 1 | solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1092",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "disproved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"chromatic-number",
"graph-theory"
] | The maintained Erdős Problems source lists problem 1092 as “disproved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1092",
"label": "Erdős Problems #1092: disproved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://h... | 2026-09-27T00:00:00 | Erdős Problems #1092: disproved (Lean): https://www.erdosproblems.com/1092
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | https://www.erdosproblems.com/1092 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,483 | EP-1093 | Erdős Problem #1093 | For $n\geq 2k$ we define the deficiency of $\binom{n}{k}$ as follows. If $\binom{n}{k}$ is divisible by a prime $p\leq k$ then the deficiency is undefined. Otherwise, the deficiency is the number of $0\leq i<k$ such that $n-i$ is $k$-smooth, that is, divisible only by primes $\leq k$.
Are there infinitely many binomia... | Erdős Problems (#1093, https://www.erdosproblems.com/1093). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1093). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1093",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"binomial-coefficients"
] | null | null | null | null | https://www.erdosproblems.com/1093 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,484 | EP-1094 | Erdős Problem #1094 | For all $n\geq 2k$ the least prime factor of $\binom{n}{k}$ is $\leq \max(n/k,k)$, with only finitely many exceptions. | Erdős Problems (#1094, https://www.erdosproblems.com/1094). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1094). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1094",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"binomial-coefficients"
] | null | null | null | null | https://www.erdosproblems.com/1094 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,485 | EP-1095 | Erdős Problem #1095 | Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$. Estimate $g(k)$. | Erdős Problems (#1095, https://www.erdosproblems.com/1095). Maintainer status: OPEN. OEIS: A003458. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1095). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1095",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"binomial-coefficients"
] | null | null | null | null | https://www.erdosproblems.com/1095 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,486 | EP-1096 | Erdős Problem #1096 | Let $1<q<1+\epsilon$ and consider the set of numbers of the shape $\sum_{i\in S}q^i$ (for all finite $S$), ordered by size as $0=x_1<x_2<\cdots$.
Is it true that, provided $\epsilon>0$ is sufficiently small, $x_{k+1}-x_k \to 0$? | Erdős Problems (#1096, https://www.erdosproblems.com/1096). Maintainer status: PROVED (LEAN). Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1096). | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1096",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | The maintained Erdős Problems source lists problem 1096 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1096",
"label": "Erdős Problems #1096: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://hugg... | 2026-09-27T00:00:00 | Erdős Problems #1096: proved (Lean): https://www.erdosproblems.com/1096
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | https://www.erdosproblems.com/1096 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,487 | EP-1097 | Erdős Problem #1097 | Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$?
In particular, are there always $O(n^{3/2})$ many such $d$? | Erdős Problems (#1097, https://www.erdosproblems.com/1097). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1097). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1097",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"additive-combinatorics"
] | null | null | null | null | https://www.erdosproblems.com/1097 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,488 | EP-1100 | Erdős Problem #1100 | If $1=d_1<\cdots<d_{\tau(n)}=n$ are the divisors of $n$, then let $\tau_\perp(n)$ count the number of $i$ for which $(d_i,d_{i+1})=1$.
Is it true that $\tau_\perp(n)/\omega(n)\to \infty$ for almost all $n$? Is it true that\[\tau_\perp(n)< \exp((\log n)^{o(1)})\]for all $n$?
Let\[g(k) = \max_{\omega(n)=k}\tau_\perp(n)... | Erdős Problems (#1100, https://www.erdosproblems.com/1100). Maintainer status: OPEN. OEIS: A325864. | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1100",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"divisors"
] | null | null | null | null | https://www.erdosproblems.com/1100 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,489 | EP-1101 | Erdős Problem #1101 | If $u=\{u_1<u_2<\cdots\}$ is a sequence of integers such that $(u_i,u_j)=1$ for all $i\neq j$ and $\sum \frac{1}{u_i}<\infty$ then let $\{a_1<a_2<\cdots\}$ be the sequence of integers which are not divisible by any of the $u_i$. For any $x$ define $t_x$ by\[u_1\cdots u_{t_x}\leq x< u_1\cdots u_{t_x}u_{t_x+1}.\]We call ... | Erdős Problems (#1101, https://www.erdosproblems.com/1101). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1101). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1101",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1101 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,490 | EP-1103 | Erdős Problem #1103 | Let $A$ be an infinite sequence of integers such that every $n\in A+A$ is squarefree. How fast must $A$ grow? | Erdős Problems (#1103, https://www.erdosproblems.com/1103). Maintainer status: OPEN. OEIS: A392164. | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1103",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1103 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,491 | EP-1104 | Erdős Problem #1104 | Let $f(n)$ be the maximum possible chromatic number of a triangle-free graph on $n$ vertices. Estimate $f(n)$. | Erdős Problems (#1104, https://www.erdosproblems.com/1104). Maintainer status: OPEN. OEIS: A292528. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1104). | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1104",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"chromatic-number"
] | null | null | null | null | https://www.erdosproblems.com/1104 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,492 | EP-1105 | Erdős Problem #1105 | The anti-Ramsey number $\mathrm{AR}(n,G)$ is the maximum possible number of colours in which the edges of $K_n$ can be coloured without creating a rainbow copy of $G$ (i.e. one in which all edges have different colours).
Let $C_k$ be the cycle on $k$ vertices. Is it true that\[\mathrm{AR}(n,C_k)=\left(\frac{k-2}{2}+\f... | Erdős Problems (#1105, https://www.erdosproblems.com/1105). Maintainer status: PROVED. OEIS: A399683, A399687. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1105). | 1 | solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1105",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | null | [
"erdos-problems",
"graph-theory",
"ramsey-theory"
] | The maintained Erdős Problems source lists problem 1105 as “proved” on 2026-09-27. No Lean verification is inferred from the existence of a formalized statement.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1105",
"label": "Erdős Problems #1105: proved"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingface... | 2026-09-27T00:00:00 | Erdős Problems #1105: proved: https://www.erdosproblems.com/1105
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | https://www.erdosproblems.com/1105 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,493 | EP-1106 | Erdős Problem #1106 | Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of\[\prod_{1\leq k\leq n}p(k).\]Does $F(n)\to \infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$? | Erdős Problems (#1106, https://www.erdosproblems.com/1106). Maintainer status: OPEN. OEIS: A194259, A194260. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1106). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1106",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1106 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,494 | EP-1107 | Erdős Problem #1107 | Let $r\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\mid n$. Is every large integer the sum of at most $r+1$ many $r$-powerful numbers? | Erdős Problems (#1107, https://www.erdosproblems.com/1107). Maintainer status: OPEN. OEIS: A056828, A392342, A392343. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1107). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1107",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"powerful"
] | null | null | null | null | https://www.erdosproblems.com/1107 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,495 | EP-1108 | Erdős Problem #1108 | Let\[A = \left\{ \sum_{n\in S}n!: S\subset \mathbb{N}\textrm{ finite}\right\}.\]If $k\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitely many powerful numbers? | Erdős Problems (#1108, https://www.erdosproblems.com/1108). Maintainer status: OPEN. OEIS: A051761, A115645, A025494. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1108). | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1108",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory",
"factorials"
] | null | null | null | null | https://www.erdosproblems.com/1108 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,496 | EP-1109 | Erdős Problem #1109 | Let $f(N)$ be the size of the largest subset $A\subseteq \{1,\ldots,N\}$ such that every $n\in A+A$ is squarefree. Estimate $f(N)$. In particular, is it true that $f(N)\leq N^{o(1)}$, or even $f(N) \leq (\log N)^{O(1)}$? | Erdős Problems (#1109, https://www.erdosproblems.com/1109). Maintainer status: OPEN. OEIS: A392164, A392165. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1109). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1109",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1109 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,497 | EP-1110 | Erdős Problem #1110 | Let $p>q\geq 2$ be two coprime integers. We call $n$ representable if it is the sum of integers of the form $p^kq^l$, none of which divide each other.
If $\{p,q\}\neq \{2,3\}$ then what can be said about the density of non-representable numbers? Are there infinitely many coprime non-representable numbers? | Erdős Problems (#1110, https://www.erdosproblems.com/1110). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1110). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1110",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1110 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,498 | EP-1111 | Erdős Problem #1111 | If $G$ is a finite graph and $A,B$ are disjoint sets of vertices then we call $A,B$ anticomplete if there are no edges between $A$ and $B$.
If $t,c\geq 1$ then there exists $d\geq 1$ such that if $\chi(G)\geq d$ and $\omega(G)<t$ then there are anticomplete sets $A,B$ with $\chi(A)\geq \chi(B)\geq c$. | Erdős Problems (#1111, https://www.erdosproblems.com/1111). Maintainer status: OPEN. | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1111",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"graph-theory"
] | null | null | null | null | https://www.erdosproblems.com/1111 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,499 | EP-1112 | Erdős Problem #1112 | Let $1\leq d_1<d_2$ and $k\geq 3$. Does there exist an integer $r$ such that if $B=\{b_1<\cdots\}$ is a lacunary sequence of positive integers with $b_{i+1}\geq rb_i$ then there exists a sequence of positive integers $A=\{a_1<\cdots\}$ such that\[d_1\leq a_{i+1}-a_i\leq d_2\]for all $i\geq 1$ and $(kA)\cap B=\emptyset$... | Erdős Problems (#1112, https://www.erdosproblems.com/1112). Maintainer status: OPEN (LEAN). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1112",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "unrefereed_claim",
"maintainer_status": "open (Lean)"
} | null | null | null | null | [
"erdos-problems",
"additive-combinatorics"
] | null | null | null | null | https://www.erdosproblems.com/1112 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,500 | EP-1113 | Erdős Problem #1113 | A positive odd integer $m$ such that none of $2^km+1$ are prime for $k\geq 0$ is called a Sierpinski number. We say that a set of primes $P$ is a covering set for $m$ if every $2^km+1$ is divisible by some $p\in P$.
Are there Sierpinski numbers with no finite covering set of primes? | Erdős Problems (#1113, https://www.erdosproblems.com/1113). Maintainer status: OPEN. Sierpinski numbers OEIS: A076336. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1113). | 1 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1113",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | Sierpinski numbers | [
"erdos-problems",
"number-theory",
"covering-systems"
] | null | null | null | null | https://www.erdosproblems.com/1113 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,501 | EP-1117 | Erdős Problem #1117 | Let $f(z)$ be an entire function which is not a monomial. Let $\nu(r)$ count the number of $z$ with $\lvert z\rvert=r$ such that $\lvert f(z)\rvert=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$. (This is a finite quantity if $f$ is not a monomial.)
Is it possible for\[\limsup \nu(r)=\infty?\]Is it possible for\[\liminf \n... | Erdős Problems (#1117, https://www.erdosproblems.com/1117). Maintainer status: OPEN. | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1117",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"analysis"
] | null | null | null | null | https://www.erdosproblems.com/1117 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,502 | EP-1120 | Erdős Problem #1120 | Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$. Let\[E= \{ z: \lvert f(z)\rvert \leq 1\}.\]What is the shortest length of a path in $E$ joining $z=0$ to $\lvert z\rvert =1$? | Erdős Problems (#1120, https://www.erdosproblems.com/1120). Maintainer status: OPEN. | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1120",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"analysis"
] | null | null | null | null | https://www.erdosproblems.com/1120 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,503 | EP-1122 | Erdős Problem #1122 | Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let\[A=\{ n \geq 1: f(n+1)< f(n)\}.\]If $\lvert A\cap [1,X]\rvert =o(X)$ then must $f(n)=c\log n$ for some $c\in \mathbb{R}$? | Erdős Problems (#1122, https://www.erdosproblems.com/1122). Maintainer status: OPEN. | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1122",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"number-theory"
] | null | null | null | null | https://www.erdosproblems.com/1122 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,504 | EP-1129 | Erdős Problem #1129 | For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.
Describe which choice of $x_i$ minimise\[\Lambda(x_1,\ldots,x_n)=\max_{x\in [-1,1]} \sum_k \lvert l_k(x)\rvert.\] | Erdős Problems (#1129, https://www.erdosproblems.com/1129). Maintainer status: PROVED (LEAN). | 1 | solved | null | null | 6 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1129",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved (Lean)"
} | null | null | null | null | [
"erdos-problems",
"analysis",
"polynomials"
] | The maintained Erdős Problems source lists problem 1129 as “proved (Lean)” on 2026-09-27. The source marks a Lean verification of the resolution.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1129",
"label": "Erdős Problems #1129: proved (Lean)"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://hugg... | 2026-09-27T00:00:00 | Erdős Problems #1129: proved (Lean): https://www.erdosproblems.com/1129
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | https://www.erdosproblems.com/1129 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,505 | EP-1130 | Erdős Problem #1130 | For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.
Let $x_0=-1$ and $x_{n+1}=1$ and\[\Upsilon(x_1,\ldots,x_n)=\min_{0\leq i\leq n}\max_{x\in[x_i,x_{i+1}]} \sum_k \lvert l_k(x)\rvert.\]Is it true that\[\Up... | Erdős Problems (#1130, https://www.erdosproblems.com/1130). Maintainer status: PROVED. | 1 | solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1130",
"checked_at": "2026-10-16",
"kind": "resolution",
"evidence": "maintained_source",
"maintainer_status": "proved"
} | null | null | null | null | [
"erdos-problems",
"analysis",
"polynomials"
] | The maintained Erdős Problems source lists problem 1130 as “proved” on 2026-09-27. No Lean verification is inferred from the existence of a formalized statement.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://www.erdosproblems.com/1130",
"label": "Erdős Problems #1130: proved"
},
{
"url": "https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml",
"label": "Erdős Problems machine-readable status list (snapshot checked 2026-09-27)"
},
{
"url": "https://huggingface... | 2026-09-27T00:00:00 | Erdős Problems #1130: proved: https://www.erdosproblems.com/1130
Erdős Problems machine-readable status list (snapshot checked 2026-09-27): https://github.com/teorth/erdosproblems/blob/main/data/problems.yaml | https://www.erdosproblems.com/1130 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,506 | EP-1131 | Erdős Problem #1131 | For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.
What is the minimal value of\[I(x_1,\ldots,x_n)=\int_{-1}^1 \sum_k \lvert l_k(x)\rvert^2\mathrm{d}x?\]In particular, is it true that\[\min I =2-(1+o(1))\... | Erdős Problems (#1131, https://www.erdosproblems.com/1131). Maintainer status: OPEN. | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1131",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"analysis",
"polynomials"
] | null | null | null | null | https://www.erdosproblems.com/1131 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,507 | EP-1132 | Erdős Problem #1132 | For $x_1,\ldots,x_n\in [-1,1]$ let\[l_k(x)=\frac{\prod_{i\neq k}(x-x_i)}{\prod_{i\neq k}(x_k-x_i)},\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\neq k$.
Let $x_1,x_2,\ldots\in [-1,1]$ be an infinite sequence, and let\[L_n(x) = \sum_{1\leq k\leq n}\lvert l_k(x)\rvert,\]where each $l_k(x)$ is defined above ... | Erdős Problems (#1132, https://www.erdosproblems.com/1132). Maintainer status: OPEN. | 1 | partially_solved | null | null | 2 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1132",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"analysis",
"polynomials"
] | null | null | null | null | https://www.erdosproblems.com/1132 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,508 | EP-1133 | Erdős Problem #1133 | Let $C>0$. There exists $\epsilon>0$ such that if $n$ is sufficiently large the following holds.
For any $x_1,\ldots,x_n\in [-1,1]$ there exist $y_1,\ldots,y_n\in [-1,1]$ such that, if $P$ is a polynomial of degree $m<(1+\epsilon)n$ with $P(x_i)=y_i$ for at least $(1-\epsilon)n$ many $1\leq i\leq n$, then\[\max_{x\in ... | Erdős Problems (#1133, https://www.erdosproblems.com/1133). Maintainer status: OPEN. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1133). | 1 | partially_solved | null | null | 3 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | null | {
"source_url": "https://www.erdosproblems.com/1133",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | null | [
"erdos-problems",
"analysis",
"polynomials"
] | null | null | null | null | https://www.erdosproblems.com/1133 | Erdős Problems (erdosproblems.com) | null | [] | null | null | null | null | null | null | null | null | null | null |
2,509 | EP-1135 | Erdős Problem #1135 | Define $f:\mathbb{N}\to \mathbb{N}$ by $f(n)=n/2$ if $n$ is even and $f(n)=\frac{3n+1}{2}$ if $n$ is odd.
Given any integer $m\geq 1$ does there exist $k\geq 1$ such that $f^{(k)}(m)=1$? | Erdős Problems (#1135, https://www.erdosproblems.com/1135). Maintainer status: OPEN. Erdős prize: $500. Collatz conjecture OEIS: A006370, A008908. Formalized in Lean (google-deepmind/formal-conjectures, ErdosProblems.1135). | 5 | partially_solved | null | null | 1 | 8 | 0 | 0 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 8,
"name": "erdos_problems",
"display_name": "Erdős Problems",
"description": "A collection of open problems posed by Paul Erdős, one of the most prolific mathematicians of the 20th century. These problems span combinatorics, number theory, graph theory, and analysis.",
"slug": "erdos-problems",
"or... | [
"NT-002",
"OPG-432"
] | {
"source_url": "https://www.erdosproblems.com/1135",
"checked_at": "2026-10-16",
"kind": "partial",
"evidence": "maintained_source",
"maintainer_status": "open"
} | null | null | null | Collatz conjecture | [
"erdos-problems",
"number-theory",
"iterated-functions"
] | null | null | null | null | https://www.erdosproblems.com/1135 | Erdős Problems (erdosproblems.com) | $500 | [] | null | null | null | null | null | null | null | null | null | null |
2,510 | KOU-21.1 | Kourovka Notebook Problem 21.1 | Let $n$ be a positive integer. For a finite group $K$ and an automorphism $\varphi$ of $K$ of order dividing $n$, let
$$
X_{n,\varphi}(K):=\{x\in K\mid x x^{\varphi}\cdots x^{\varphi^{n-1}}=1\}.
$$
Let $c_n$ be the supremum of the ratios $|X_{n,\varphi}(H)|/|H|$ over all finite groups $H$ and their automorphisms $\va... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.1.
Discussion and literature:
This question is open even when $n\geqslant 5$ is prime. A. Abdollahi, M. S. Malekan
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classif... | 2 | partially_solved | A. Abdollahi, M. S. Malekan | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,511 | KOU-21.2 | Kourovka Notebook Problem 21.2 | Let $S$ be a finite simple group, and let $G$ be a finite group for which there exists a bijection $f:G\to S$ such that $|x|$ divides $|f(x)|$ for all $x\in G$. Must $G$ necessarily be simple? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.2.
Discussion and literature:
M. Amiri
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof that the element-ord... | 2 | open | M. Amiri | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,512 | KOU-21.3 | Kourovka Notebook Problem 21.3 | Let $G=A_n$ or $S_n$ and let $H,K$ be soluble subgroups of $G$. For all sufficiently large $n$, can we always find an element $x\in G$ such that $H\cap K^x=1$? Does this hold for all $n\geqslant 21$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.3.
Discussion and literature:
Note that the conclusion is false when $G=S_{20}$ and $H=K=(S_4\wr S_4)\times S_4$. M. Anagnostopoulou-Merkouri, T. C. Burness
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)... | 2 | partially_solved | M. Anagnostopoulou-Merkouri, T. C. Burness | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) Yes, we can (A. Rizzoli, *Preprint*, 2026, https://kourovkanotebookorg.wordpress.com/wp-con... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,513 | KOU-21.4 | Kourovka Notebook Problem 21.4 | Let $G$ be a finite group with trivial solvable radical and let $H_1,\ldots,H_5$ be solvable subgroups of $G$. Then do there always exist elements $x_i\in G$ such that $\bigcap_i H_i^{x_i}=1$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.4.
Discussion and literature:
Cf. 17.41(b). M. Anagnostopoulou-Merkouri, T. C. Burness
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current litera... | 2 | open | M. Anagnostopoulou-Merkouri, T. C. Burness | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,514 | KOU-21.5 | Kourovka Notebook Problem 21.5 | Let $p$ be a prime. Let $G$ be a transitive subgroup of the group of finitary permutations $\operatorname{FSym}(\Omega)$ of a set $\Omega$, let $N$ be a normal subgroup of $G$, and let $S$ be a transitive Sylow $p$-subgroup of $G$.
(a) Is it true that $S\cap N$ is a Sylow $p$-subgroup of $N$?
(b) Is it true that $SN/... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.5.
Discussion and literature:
A. O. Asar
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No source deciding the norm... | 2 | partially_solved | A. O. Asar | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Subproblem confirmed solved (*) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] a) Yes, it is true (AI solution confirmed by F. Leinen: \\ P. Monticone, *Preprint*, 2026,\nht... | null | null | null | [Discussion #4 literature review, 2026-10-14] **partially solved**; the dataset currently labels it open. A. O. Asar, three questions on transitive Sylow p-subgroups S of a transitive group G of finitary permutations: (a) is S ∩ N a Sylow p-subgroup of a normal subgroup N; (b) is SN/N a Sylow p-subgroup of G/N; (c) are... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,515 | KOU-21.6 | Kourovka Notebook Problem 21.6 | Let p be a prime. A totally imprimitive p-group H of finitary permutations is said to have the cyclic-block property if in the cycle decomposition of every element the support of every cycle is a block for H. Let G be a transitive subgroup of the group of finitary permutations $\operatorname{FSym}(\Omega)$ of a set $\O... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.6.
Discussion and literature:
A. O. Asar
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof or counterexample ... | 2 | open | A. O. Asar | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,516 | KOU-21.7 | Kourovka Notebook Problem 21.7 | (Well-known problem). A finite group G is called an IYB-group if it is isomorphic to the permutation group of a finite involutive non-degenerate set-theoretic solution of the Yang--Baxter equation, or equivalently, G is isomorphic to the multiplicative group of a finite left brace. Assume that the Sylow subgroups of a ... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.7.
Discussion and literature:
A. Ballester-Bolinches
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No closure theo... | 3 | open | A. Ballester-Bolinches | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,517 | KOU-21.8 | Kourovka Notebook Problem 21.8 | As in 17.57, let $r(m)=\{r+km\mid k\in\mathbb Z\}$ for integers $0\leqslant r<m$; for $r_1(m_1)\cap r_2(m_2)=\emptyset$ let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the involution which interchanges $r_1+tm_1$ and $r_2+tm_2$ for each integer t and fixes everything else, and let $\operatorname{CT}(\mathbb Z... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.8.
Discussion and literature:
V. G. Bardakov, A. L. Iskra
Yes, it is true (Junyao Pan, Preprint of 14 April 2026, https://arxiv.org/abs/2604.12553; P. Monticone, Preprint of 30 April 2026, https://kourovkanotebookorg.wordpress.com/... | 2 | solved | V. G. Bardakov, A. L. Iskra | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (Junyao Pan, *Preprint of 14 April 2026*, https://arxiv.org/abs/2604.12553; \\ AI soluti... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,518 | KOU-21.9 | Kourovka Notebook Problem 21.9 | Let $F$ be a non-abelian free pro-$p$ group of finite rank. Can one find a finite collection $U_1,\ldots,U_n$ of open subgroups of $F$, including $F$ itself, such that the only subgroup of $F$ which is contained in $U_i$ and is characteristic in $U_i$ for every $i$ is the trivial subgroup? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.9.
Discussion and literature:
Y. Barnea
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No finite characteristic-ope... | 2 | open | Y. Barnea | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,519 | KOU-21.10 | Kourovka Notebook Problem 21.10 | We call a group presentation finite if it represents a finite group. We say that a presentation is just finite if it is finite and is no longer finite on removal of any relation from it. Is it true that every finite group has a just finite presentation? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.10.
Discussion and literature:
Note that if a group has a balanced presentation, then it is just finite. A similar argument can be applied for some p-groups using the Golod--Shafarevich inequality. Y. Barnea
<!-- LITERATURE-TRIAGE:... | 2 | solved | Y. Barnea | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-09-27T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (M. Lackenby, *Preprint*, 2026, https://arxiv.org/abs/2605.10402)."
} | null | null | null | null | null | Every finite group admits a just finite presentation. The maintained notebook marks this problem solved. Some cited solutions are preprints or letters; notebook acceptance does not imply journal peer review.
**What remains.** No unresolved part of the stated question is identified in the cited result. | [
{
"url": "https://arxiv.org/abs/1401.0300v46",
"label": "The Kourovka Notebook, 21st edition, version 46 (2026-09-27)"
},
{
"url": "https://arxiv.org/abs/2605.10402",
"label": "M. Lackenby, solution cited in the notebook (2026)"
},
{
"url": "https://huggingface.co/datasets/ulamai/Unsolve... | 2026-09-27T00:00:00 | The Kourovka Notebook, 21st edition, version 46 (2026-09-27): https://arxiv.org/abs/1401.0300v46
M. Lackenby, solution cited in the notebook (2026): https://arxiv.org/abs/2605.10402 | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,520 | KOU-21.11 | Kourovka Notebook Problem 21.11 | Can some or all groups of the following sorts be written as homomorphic images of nonprincipal ultraproducts of countable families of groups? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.11.
Discussion and literature:
This is Question 18 in (G. M. Bergman, Pacific J. Math., 274 (2015) 451--495).
(a) Infinite finitely generated groups of finite exponent.
(b) For an infinite set X, the group of those permutations of... | 2 | partially_solved | G. M. Bergman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,521 | KOU-21.12 | Kourovka Notebook Problem 21.12 | Suppose that $U$ is a nonprincipal ultrafilter on $\omega$, and $B$ is a group such that every element $b\in B$ belongs to a subgroup of $B$ that is a homomorphic image of $\mathbb Z^\omega/U$. Must $B$ then be a homomorphic image of an ultraproduct group $(\prod_{i\in\omega}G_i)/U$ for some groups $G_i$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.12.
Discussion and literature:
This is Question 19 in (G. M. Bergman, Pacific J. Math., 274 (2015) 451--495).
An affirmative answer would imply that every torsion group was such a homomorphic image for every $U$, and so would give ... | 2 | solved | G. M. Bergman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it need not (S. M. Corson, *J. Algebra*, **681** (2025), 306--317)."
} | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,522 | KOU-21.13 | Kourovka Notebook Problem 21.13 | Does $\mathbb Z^\omega$ have a subgroup whose dual is free abelian of still larger rank (the largest possible being $2^{2^{\aleph_0}}$)? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.13.
Discussion and literature:
It is known that the group $\mathbb Z^\omega$ has a subgroup whose dual is free abelian of rank $2^{\aleph_0}$ (see 17.24 in Archive).
This is Question 11 in (G. M. Bergman, Portugaliae Math., 69 (201... | 2 | partially_solved | G. M. Bergman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,523 | KOU-21.14 | Kourovka Notebook Problem 21.14 | Suppose $\alpha$ is an endomorphism of a group G such that for every group H and every homomorphism $f:G\to H$, there exists an endomorphism $\beta_f$ of $H$ such that $\beta_f f=f\alpha$. Must $\alpha$ then be either an inner automorphism of G or the trivial endomorphism? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.14.
Discussion and literature:
This is Question 5 in (G. M. Bergman, Publ. Matem., 56 (2012), 91--126). G. M. Bergman
Yes, it must (F. Fournier-Facio, Preprint, 2026, https://arxiv.org/abs/2604.05728)
<!-- LITERATURE-TRIAGE:BEGIN ... | 2 | solved | G. M. Bergman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it must (F. Fournier-Facio, *Preprint*, 2026,\nhttps://arxiv.org/abs/2604.05728)"
} | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,524 | KOU-21.15 | Kourovka Notebook Problem 21.15 | Suppose B is a subgroup of the symmetric group $S_\Omega$ on an infinite set $\Omega$. Will the amalgamated free product $S_\Omega *_B S_\Omega$ of two copies of $S_\Omega$ with amalgamation of B be embeddable in $S_\Omega$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.15.
Discussion and literature:
This is a weakened form of the group case of Question 4.4 in (G. M. Bergman, Indag. Math., 18 (2007), 349--403).
It is known that $S_\Omega *_B S_\Omega$ need not be so embeddable by a map respecting ... | 2 | solved | G. M. Bergman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not necessarily: take $\\Omega$ countably infinite, $B$ a subgroup of $S_{\\Omega}$ which is not Bor... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,525 | KOU-21.16 | Kourovka Notebook Problem 21.16 | Let the width of a group (respectively, a monoid) H with respect to a generating set X mean the supremum over h $\in$ H of the least length of a group word (respectively, a monoid word) in elements of X expressing h. A group (or monoid) is said to have finite width if its width with respect to every generating set is f... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.16.
Discussion and literature:
This is Question 9 in (G. M. Bergman, Bull. London Math. Soc., 38 (2006), 429--440). G. M. Bergman
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Clas... | 2 | open | G. M. Bergman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,526 | KOU-21.17 | Kourovka Notebook Problem 21.17 | If $X$ is a class of groups, let $H(X)$ denote the class of homomorphic images of groups in $X$, let $S(X)$ denote the class of groups isomorphic to subgroups of groups in $X$, let $P(X)$ denote the class of groups isomorphic to (unrestricted) direct products of families of groups in $X$, and let $P_f(X)$ denote the cl... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.17.
Discussion and literature:
This is Question 27 in (G. M. Bergman, Algebra Universalis, 26 (1989), 267--283). G. M. Bergman
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classif... | 2 | open | G. M. Bergman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,527 | KOU-21.18 | Kourovka Notebook Problem 21.18 | Suppose that G is a finite group, and $A_1,A_2,A_3$ are subsets of G such that the multiplication map $A_1\times A_2\times A_3\to G$ is bijective. Must the subgroup $\langle A_2\rangle$ generated by $A_2$ have order divisible by the cardinality $|A_2|$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.18.
Discussion and literature:
This is Question 8 in (G. M. Bergman, J. Iranian Math. Soc., 1 (2020), 157--161).
It is known (ibid.) that the corresponding statement is true for the subgroups $\langle A_1\rangle$ and $\langle A_3\r... | 2 | solved | G. M. Bergman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, it need not (M. I. Kabenyuk, *Preprint*, 2021, https://arxiv.org/abs/2102.08605)."
} | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,528 | KOU-21.19 | Kourovka Notebook Problem 21.19 | Suppose that S and M are groups of finite Morley rank, S is an infinite group, and M is a non-trivial connected group definably and faithfully acting on S. This action is said to be irreducible if M does not leave invariant any definable non-trivial proper subgroup of S. Prove that if S is a simple group such that ever... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.19.
Discussion and literature:
A. V. Borovik
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The finite-Morley-rank ... | 2 | open | A. V. Borovik | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,529 | KOU-21.20 | Kourovka Notebook Problem 21.20 | Prove that a simple group of finite Morley rank without involutions cannot act definably, faithfully, and irreducibly on a connected group other than on itself acting by conjugation. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.20.
Discussion and literature:
A. V. Borovik
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The involution-free fin... | 2 | open | A. V. Borovik | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,530 | KOU-21.21 | Kourovka Notebook Problem 21.21 | Prove that a simple (that is, without proper non-trivial connected normal subgroups) algebraic group M over an algebraically closed field cannot act definably, faithfully, and irreducibly on a simple group of finite Morley rank other than M/Z(M). | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.21.
Discussion and literature:
A. V. Borovik
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** The algebraic-group-on-... | 2 | open | A. V. Borovik | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,531 | KOU-21.22 | Kourovka Notebook Problem 21.22 | Is the (standard, restricted) wreath product $G\wr H$ of two finitely generated Hopfian groups Hopfian? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.22.
Discussion and literature:
The same question where G is assumed to be abelian or nilpotent is equivalent to Kaplansky's direct finiteness conjecture; see (H. Bradford, F. Fournier-Facio, Math. Z., 308, no. 4 (2024), Paper no. 58... | 2 | partially_solved | H. Bradford, F. Fournier-Facio | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,532 | KOU-21.23 | Kourovka Notebook Problem 21.23 | A graph is called a cograph if it has no induced subgraph isomorphic to a path with 4 vertices. A graph is said to be chordal if it has no induced cycles with n vertices for every $n\geqslant 4$. For a finite group G, the enhanced power graph E(G) is the graph with vertex set G and edges \{x, y\} for all $x\ne y\in G$ ... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.23.
Discussion and literature:
In (Preprint, 2025, https://arxiv.org/abs/2510.18073) we proved that if the enhanced power graph of a given finite group is a cograph, then it is also chordal. Also the finite nonabelian simple groups ... | 2 | partially_solved | D. Bubboloni, F. Fumagalli, C. E. Praeger | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,533 | KOU-21.24 | Kourovka Notebook Problem 21.24 | For a finite group G, the power graph P(G) is the graph with vertex set G and edges \{x, y\} for all $x\ne y\in G$ such that either $x\in\langle y\rangle$ or $y\in\langle x\rangle$. Is it true that, for every finite group G, if P(G) is a cograph, then P(G) is chordal? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.24.
Discussion and literature:
Cf. 21.23. This holds if every element of G has prime power order (D. Bubboloni, F. Fumagalli, C. E. Praeger, Preprint, 2025, https://arxiv.org/abs/2510.18073) and if G is a nonabelian simple group (J.... | 2 | solved | D. Bubboloni, F. Fumagalli, C. E. Praeger | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is true (M. Rundstr\\\"om, *Preprint of 30 January 2026*, https://kourovkanotebookorg.wordpress.... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,534 | KOU-21.25 | Kourovka Notebook Problem 21.25 | Let $G$ be a finite simple group and let $p_1,p_2$ be any (not necessarily distinct) prime divisors of $|G|$. Then can we always find Sylow $p_i$-subgroups $H_i$ such that $G=\langle H_1,H_2\rangle$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.25.
Discussion and literature:
(T. Breuer, R. M. Guralnick).
T. C. Burness
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current ... | 2 | partially_solved | T. C. Burness | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,535 | KOU-21.26 | Kourovka Notebook Problem 21.26 | Let $G$ be a non-trivial finite group and let $p_1,\ldots,p_k$ be the distinct prime divisors of $|G|$. For each $i$, let $H_i$ be a Sylow $p_i$-subgroup of $G$. Is it true that there exists an element $x\in G$ such that for all $i$ the subgroup $H_i\cap H_i^x$ is inclusion-minimal in $\{H_i\cap H_i^g:g\in G\}$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.26.
Discussion and literature:
(F. Lisi, L. Sabatini).
T. C. Burness
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current litera... | 2 | partially_solved | T. C. Burness | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,536 | KOU-21.27 | Kourovka Notebook Problem 21.27 | A permutation on a set $\Omega$ is called a derangement if it has no fixed points in $\Omega$. Let G be a finite simple transitive permutation group. Is it true that every element in G is the product of two derangements? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.27.
Discussion and literature:
(M. Larsen, A. Shalev, P. H. Tiep).
T. C. Burness
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Cu... | 2 | partially_solved | T. C. Burness | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,537 | KOU-21.28 | Kourovka Notebook Problem 21.28 | Let $G$ be a finite simple transitive permutation group, and let $\delta(G)$ be the proportion of derangements in $G$. Is it true that $\delta(G)\geqslant 89/325$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.28.
Discussion and literature:
Note that $\delta(G)=89/325$ for the action of the Tits group $G={}^2F_4(2)'$ on the cosets of a maximal parabolic subgroup of the form $2^2.[2^8].S_3$ (Forum Math. Sigma, 13 (2025), paper no. e98, 62 ... | 2 | partially_solved | T. C. Burness, M. Fusari | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,538 | KOU-21.29 | Kourovka Notebook Problem 21.29 | Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite primitive permutation group with a regular suborbit (that is, $G$ has a trivial 2-point stabiliser). Then is it true that for all $\alpha,\beta\in\Omega$, there exists $\gamma\in\Omega$ such that the 2-point stabilisers $G_{\alpha,\gamma}$ and $G_{\beta,\gamma}$ ar... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.29.
Discussion and literature:
T. C. Burness, M. Giudici
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessme... | 2 | solved | T. C. Burness, M. Giudici | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not always (A. Rizzoli, A. R. Thomas, *Preprint*, 2026, https://arxiv.org/abs/2609.01367)."
} | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (no)**; the dataset currently labels it partially solved. T. C. Burness, M. Giudici: for a finite primitive group with a regular suborbit, is there for all α, β a point γ with G_{α,γ} and G_{β,γ} both trivial? Notebook v48: ∗21.29, "No, not always", with a referenc... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,539 | KOU-21.30 | Kourovka Notebook Problem 21.30 | (Well-known question). A discrete group G is said to have the Haagerup property (also known as Gromov's a-T-menability property) if there exists a metrically proper isometric action of G on a (possibly infinite-dimensional) Hilbert space. Are all 1-relator groups Haagerup groups? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.30.
Discussion and literature:
J. O. Button
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The Haa... | 3 | partially_solved | J. O. Button | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,540 | KOU-21.31 | Kourovka Notebook Problem 21.31 | Conjecture: If N is a finite soluble group, then any regular subgroup in the holomorph Hol(N) of N is also soluble. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.31.
Discussion and literature:
N. Byott
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** No insolubl... | 3 | partially_solved | N. Byott | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,541 | KOU-21.32 | Kourovka Notebook Problem 21.32 | Is the following problem decidable, and if so, what is its complexity? Given a finite group G, is there a finite group H such that the derived subgroup of H is isomorphic to G? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.32.
Discussion and literature:
P. J. Cameron
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No decision procedure o... | 2 | solved | P. J. Cameron | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it is decidable, in nondeterministic quasi-polynomial time (S. Kalithasan, V. Z. Thomas, *Preprint*... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved according to the notebook (decidable)**, or partially solved if the question is read as asking for the exact complexity; the dataset currently labels it open. P. J. Cameron: is it decidable whether a finite group G is the derived subgroup of some finite group, and ... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,542 | KOU-21.33 | Kourovka Notebook Problem 21.33 | Does an analogue of Dunwoody's theorem hold for totally disconnected locally compact groups, that is, must a tdlc group of rational discrete cohomological dimension at most 1 be topologically isomorphic to the fundamental group of a graph of profinite groups? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.33.
Discussion and literature:
I. Castellano
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No tdlc analogue of Dun... | 2 | open | I. Castellano | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,543 | KOU-21.34 | Kourovka Notebook Problem 21.34 | (Well-known problem). A group $G$ is a unique product group if, for any nonempty finite subsets $A,B$ of $G$, there exists an element of $G$ which can be written uniquely as $ab$ with $a\in A$ and $b\in B$. A group $G$ is locally invariant orderable if $G$ admits a partial order $<$ such that for all $g,h\in G$ with $h... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.34.
Discussion and literature:
A. Clay
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No unique-product group faili... | 3 | open | A. Clay | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,544 | KOU-21.35 | Kourovka Notebook Problem 21.35 | Let $G$ be a finite group, $w$ a multilinear commutator group-word, and $p$ a prime. Suppose that $p$ divides the order $|xy|$ whenever $x$ is a $w$-value of $p'$-order in $G$ and $y$ is a $w$-value in $G$ of order divisible by $p$. Is it true that then the verbal subgroup $w(G)$ must be $p$-nilpotent? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.35.
Discussion and literature:
Without the assumption that $w$ be multilinear, the answer is negative. An affirmative answer has been obtained in several special cases (J. Algebra, 609 (2022), 926--936). Y. Contreras Rojas, V. Grazi... | 2 | partially_solved | Y. Contreras Rojas, V. Grazian, C. Monetta | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "partially_solved",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it is true (unverified AI solution: Nilradical, *Preprint*, 2026... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,545 | KOU-21.36 | Kourovka Notebook Problem 21.36 | Let a hierarchy of tdlc groups $\mathbf H\mathcal K$ be defined analogously to Kropholler's hierarchy in 15.45, with $\mathcal K$ being the class of profinite groups and with the cell stabilisers of the admissible action required to be open. Is it true that $\mathbf H\mathcal K$ is closed under profinite extensions, th... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.36.
Discussion and literature:
Kropholler's hierarchy (see 15.45) is closed under finite extensions, that is, $(\mathbf H_\alpha\mathcal F)\mathcal F\subseteq \mathbf H_\alpha\mathcal F$ for every $\alpha$ (P. Kropholler, J. Pure Ap... | 2 | partially_solved | G. C. Cook | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,546 | KOU-21.37 | Kourovka Notebook Problem 21.37 | By definition, a constructible totally disconnected, locally compact (tdlc) group is the result of a sequence of profinite extensions and ascending HNN-extensions starting from the trivial group. Are soluble tdlc groups of type $FP_\infty$ constructible? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.37.
Discussion and literature:
As in the discrete case, soluble constructible tdlc groups have type $FP_\infty$ (G. C. Cook, I. Castellano, J. Algebra, 543 (2020), 54--97).
G. C. Cook
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature... | 2 | partially_solved | G. C. Cook | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,547 | KOU-21.38 | Kourovka Notebook Problem 21.38 | The spread of a group $G$ is the greatest nonnegative integer $k$ such that for all nontrivial elements $x_1,\ldots,x_k\in G$ there exists $y\in G$ such that $\langle x_1,y\rangle=\cdots=\langle x_k,y\rangle=G$, or is $\infty$ in case there is no such maximum. Does there exist a group with spread equal to 1? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.38.
Discussion and literature:
(S. Harper, C. Donoven).
Such a group must be infinite if it exists (T. C. Burness, R. M. Guralnick, S. Harper, Ann. Math., 193 (2021), 619--687). S. Corson
<!-- LITERATURE-TRIAGE:BEGIN -->
## Litera... | 2 | solved | S. Corson | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, such a group exists (AI solution confirmed by S. Corson: \\ Nilradical, *Preprint*, 2026, https://k... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (yes)**; the dataset currently labels it partially solved. S. Harper, C. Donoven (submitted by S. Corson): does there exist a group with spread equal to 1? Notebook v48: ∗21.38, "Yes, such a group exists", AI solution confirmed by S. Corson (Nilradical, note of 17 ... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,548 | KOU-21.39 | Kourovka Notebook Problem 21.39 | Are there any locally finite, characteristically simple groups with finitely many orbits under automorphisms that are not residually finite? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.39.
Discussion and literature:
It is known that there exist residually finite, locally finite, characteristically simple groups with finitely many orbits under automorphisms (A. B. Apps, J. Algebra, 81 (1983), 320--339).
A. Dantas,... | 2 | partially_solved | A. Dantas, E. de Melo | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,549 | KOU-21.40 | Kourovka Notebook Problem 21.40 | Let G be a subgroup of GL(n, Q) with finitely many orbits under automorphisms. Is G a virtually soluble group? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.40.
Discussion and literature:
A. Dantas, E. de Melo
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No theorem that... | 2 | open | A. Dantas, E. de Melo | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "unrefereed_claim",
"canonical_status": "open",
"checked_at": "2026-10-16",
"notes": "Marked with # (unverified AI solution claim) in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Unverified AI claim] Yes, it is (unverified AI solutions: Achyuth Jayadevan, *Preprint of 14 September... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,550 | KOU-21.41 | Kourovka Notebook Problem 21.41 | A group is said to be self-similar if it admits a faithful state-closed representation by automorphisms of a regular one-rooted m-tree for some m. Can a torsion-free finitely presented metabelian group which is self-similar contain a subgroup isomorphic to the restricted wreath product H = $\mathbb Z\wr\mathbb Z$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.41.
Discussion and literature:
It is known that $\mathbb Z\wr\mathbb Z$ itself is self-similar (A. C. Dantas, T. M. G. Santos, S. N. Sidki, J. Algebra, 567 (2021), 564--581). A. Dantas, S. Sidki
<!-- LITERATURE-TRIAGE:BEGIN -->
## ... | 2 | partially_solved | A. Dantas, S. Sidki | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,551 | KOU-21.42 | Kourovka Notebook Problem 21.42 | Let $\mathcal T_{d,c}$ denote the class of $d$-generated, torsion-free nilpotent groups having class $c$. Are there $\mathcal T_{3,3}$-groups that are not self-similar? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.42.
Discussion and literature:
It is known that $\mathcal T_{d,2}$-groups are self-similar for all $d$, that $\mathcal T_{2,3}$-groups are self-similar, and that there are $\mathcal T_{4,3}$-groups that are not self-similar (A. Berl... | 2 | partially_solved | A. Dantas, S. Sidki | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,552 | KOU-21.43 | Kourovka Notebook Problem 21.43 | Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.43.
Discussion and literature:
M. Deaconescu
This is not always true. In a direct product of $N$ copies of $A_5$ there are $60^N$ elements, while the number of elements of order exactly 30 is $60^N-45^N-40^N-36^N+25^N+21^N+16^N-1$.... | 3 | solved | M. Deaconescu | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] This is not always true. In a direct product of $N$ copies of $A_5$ there are $60^N$ elements, while the... | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,553 | KOU-21.44 | Kourovka Notebook Problem 21.44 | Let $W_n=A_5\wr\cdots\wr A_5$ be the $n$-times iterated permutational wreath product of $A_5$ in its natural action (so $W_n$ acts on $5^n$ points), and let $W=\varprojlim W_n$ be the inverse limit (infinite iterated wreath product of $A_5$). Does $W$ contain a finitely generated dense subgroup of subexponential growth... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.44.
Discussion and literature:
S. Eberhard
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The infi... | 2 | solved | S. Eberhard | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] Yes, it does (AI solution confirmed by S. Eberhard: \\ Nilradical, *Preprint*, 2026,\n\nhttps://kourovka... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (yes)**; the dataset currently labels it partially solved. S. Eberhard: does the infinite iterated wreath product W of A₅ contain a finitely generated dense subgroup of subexponential growth? Notebook v48: ∗21.44, "Yes, it does", AI solution confirmed by S. Eberhar... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,554 | KOU-21.45 | Kourovka Notebook Problem 21.45 | (Well-known problem). Does there exist a finitely presented (infinite) simple group requiring more than two generators? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.45.
Discussion and literature:
Cf. 6.44 in Archive. F. Fournier-Facio
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current litera... | 3 | partially_solved | F. Fournier-Facio | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,555 | KOU-21.46 | Kourovka Notebook Problem 21.46 | (Well-known problem). Does there exist a finitely presented (infinite) simple group of finite cohomological dimension greater than 2? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.46.
Discussion and literature:
F. Fournier-Facio
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** Fi... | 3 | partially_solved | F. Fournier-Facio | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,556 | KOU-21.47 | Kourovka Notebook Problem 21.47 | (Well-known problem). Does there exist a finitely presented group $G$ such that $G\cong G\times H$ for some non-trivial group $H$? | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.47.
Discussion and literature:
The first finitely generated example was constructed in (J. M. Tyrer Jones, J. Austral. Math. Soc., 17 (1974), 174--196).
A finitely presented group that surjects onto its own direct square was constr... | 3 | partially_solved | F. Fournier-Facio | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,557 | KOU-21.48 | Kourovka Notebook Problem 21.48 | A quasimorphism on a group $G$ is a function $f:G\to\mathbb R$ such that the quantity $\sup_{g,h}|f(g)+f(h)-f(gh)|$ is finite. A quasimorphism is homogeneous if it restricts to a homomorphism on every cyclic subgroup of $G$. Let $G$ be a group admitting an unbounded homogeneous quasimorphism $G\to\mathbb R$ that is not... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.48.
Discussion and literature:
F. Fournier-Facio
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** No... | 2 | partially_solved | F. Fournier-Facio | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,558 | KOU-21.49 | Kourovka Notebook Problem 21.49 | An isometric action of a group G on a metric space S is called acylindrical if for every $\varepsilon>0$ there exist R, N > 0 such that for every two points x, y with $d(x,y)\geqslant R$, there are at most N elements $g\in G$ satisfying $d(x,gx)\leqslant\varepsilon$ and $d(y,gy)\leqslant\varepsilon$. A group is said to... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.49.
Discussion and literature:
A. Genevois
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
**Classification:** PARTIAL-PROGRESS
**Current literature assessment.** The auto... | 2 | partially_solved | A. Genevois | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,559 | KOU-21.50 | Kourovka Notebook Problem 21.50 | Does every finite 3-group $T$ have a nontrivial characteristic subgroup $C$ such that if $T$ is a Sylow 3-subgroup of a finite group $G$, then $T\cap G'=T\cap H'$, where $H=N_G(C)$?
The group $S_4$ shows that no such characteristic subgroups can be found in some Sylow 2-subgroups. | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.50.
Discussion and literature:
Such a characteristic subgroup is known to exist in $p$-groups for $p\geqslant 5$ (G. Glauberman, Math. Z., 117 (1970), 46--56), and for $p=3$ there are two characteristic subgroups $K_1,K_2$ such that... | 2 | partially_solved | G. Glauberman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,560 | KOU-21.51 | Kourovka Notebook Problem 21.51 | Let $p$ be a prime, and $P$ a finite $p$-group.
(a) Suppose that $P$ has an abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have a normal abelian subgroup of order $p^n$?
(b) Suppose that $P$ has an elementary abelian subgroup of order $p^n$. For which $n$ does $P$ necessarily have a normal elemen... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.51.
Discussion and literature:
It is easy to see that for $p=2$, the answer to (b) is "yes" only for $n=1$. The answer to both questions is "yes" for $n<(p+2)/2$ (G. G. Glauberman, J. Algebra, 319, no. 2 (2008), 800--805), as well a... | 2 | partially_solved | G. Glauberman | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,561 | KOU-21.52 | Kourovka Notebook Problem 21.52 | Let $L$ be a finite non-abelian simple group, and let $D$ be a conjugacy class of involutions in $L$. Consider the complete graph $\Gamma$ with vertex set $D$. Define an equivalence relation $\sim$ (graph coloring) on the set of edges as follows: $(a,b)\sim(c,d)$ if and only if $|ab|=|cd|$. An automorphism of the colou... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.52.
Discussion and literature:
I. B. Gorshkov
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No proof or counterexa... | 2 | open | I. B. Gorshkov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,562 | KOU-21.53 | Kourovka Notebook Problem 21.53 | In the notation of 21.52, let $\operatorname{Aut}_t(\Gamma)$ be the set of permutations $\tau\in S_D$ such that $(a,b)\sim(a^\tau,b^\tau)$ whenever $|ab|=t$ for $a,b\in D$. Clearly, $\operatorname{Aut}(\Gamma)=\bigcap_t\operatorname{Aut}_t(\Gamma)$. Is it true that for every finite simple group $G$ we have $\operatorna... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.53.
Discussion and literature:
I. B. Gorshkov
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Classification:** OPEN-TRIAGE
**Current literature assessment.** No source was found pr... | 2 | solved | I. B. Gorshkov | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | {
"evidence": "maintained_source",
"canonical_status": "solved",
"checked_at": "2026-10-16",
"notes": "Confirmed solved (*) by editors in Kourovka Notebook 21st Issue (Oct 2026, arXiv:1401.0300v48): [Solved] No, not for every (AI solution confirmed by I. B. Gorshkov: \\\n\nNilradical, *Preprint*, 2026,\nhttps:/... | null | null | null | [Discussion #4 literature review, 2026-10-14] **solved (no)**; the dataset currently labels it open. I. B. Gorshkov: for every finite simple group, is Aut(Γ) = Aut₂(Γ) ∩ Aut_p(Γ), {2, p} the two smallest prime divisors of the order? Notebook v48: ∗21.53, "No, not for every", AI solution confirmed by I. B. Gorshkov (Nil... | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
2,563 | KOU-21.54 | Kourovka Notebook Problem 21.54 | Let $G$ be a finite soluble group with triality, which means that $G$ admits a group of automorphisms $S$ isomorphic to the symmetric group of degree 3 given by the presentation $S=\langle\sigma,\rho\mid \sigma^2=\rho^3=1;\ \sigma\rho\sigma=\rho^2\rangle$ such that $m\cdot m^\rho\cdot m^{\rho^2}=1$ for all $m$ in the s... | Source: Kourovka Notebook, New Problems, 21st issue, 2026. Original problem number: 21.54.
Discussion and literature:
An affirmative answer would provide a reduction of the analogue of the Restricted Burnside Problem for Moufang loops to the nilpotent case. A. N. Grishkov, A. V. Zavarnitsine
<!-- LITERATURE-TRIAGE:BE... | 2 | open | A. N. Grishkov, A. V. Zavarnitsine | 2,026 | 17 | 10 | 0 | 0 | 2026-05-13T00:00:00 | 2026-05-13T00:00:00 | true | {
"id": 17,
"name": "group_theory",
"display_name": "Group Theory",
"description": "Problems about groups, group actions, representations, and related algebraic structures.",
"slug": "group-theory",
"order_index": 17,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 10,
"name": "kourovka_new_problems_21",
"display_name": "Kourovka Notebook (Issues 1–21)",
"description": "Unsolved and archive-solved problems in group theory from Issues 1–21 (1965–2026) of the Kourovka Notebook, edited by E. I. Khukhro and V. D. Mazurov.",
"slug": "kourovka-new-problems-21",
"ord... | null | null | null | null | null | null | null | null | null | null | null | https://arxiv.org/abs/1401.0300 | null | null | null | null | null | null | null | null | null | null | null | null | null |
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