phanerozoic/Lean4-FormalConjectures · Datasets at Hugging Face

fact stringlengths 28 3.53k	type stringclasses 8 values	library stringclasses 3 values	imports listlengths 1 7	filename stringlengths 33 87	symbolic_name stringlengths 1 87	docstring stringlengths 17 494 ⌀
getCategoryStatsMarkdown : CoreM String := do let stats ← getCategoryStats let githubSearchBaseUrl := "https://github.com/search?type=code&q=repo%3Agoogle-deepmind%2Fformal-conjectures+" return s!"\| Count \| Category \| \| ----- \| ----------------- \| \| {stats (Category.research ProblemStatus.open)} \| [Research (open)]({githubSearchBaseUrl}%22category+research+open%22)\| \| {stats (Category.research ProblemStatus.solved)} \| [Research (solved)]({githubSearchBaseUrl}%22category+research+solved%22)\| \| {stats (Category.graduate)} \| [Graduate]({githubSearchBaseUrl}%22category+graduate%22)\| \| {stats (Category.undergraduate)} \| [Undergraduate]({githubSearchBaseUrl}%22category+undergraduate%22)\| \| {stats (Category.highSchool)} \| [High School]({githubSearchBaseUrl}%22category+high_school%22)\| \| {stats (Category.API)} \| [API]({githubSearchBaseUrl}%22category+API%22)\| \| {stats (Category.test)} \| [Tests]({githubSearchBaseUrl}%22category+tests%22)\|" -- TODO(firsching): make it possible to search for subjects in doc-gen4, likely depends on -- https://github.com/google-deepmind/formal-conjectures/issues/5	def	docbuild	[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]	docbuild/scripts/overwrite_index.lean	getCategoryStatsMarkdown	null
getSubjectStatsMarkdown : CoreM String := do let tags ← getSubjectTags let mut counts : Std.HashMap AMS Nat := {} for tag in tags do for subject in tag.subjects do counts := counts.insert subject (counts.getD subject 0 + 1) let sortedCounts := counts.toArray.qsort (lt := fun (_, c1) (_, c2) => c2 < c1) let mut markdownTable := "\| Count \| AMS # \| Subject \|\n" ++ "\| ----- \| ----- \| ------- \|\n " for (subject, count) in sortedCounts do if count > 0 then let desc ← subject.getDesc let some num := subject.toNat? \| throwError "subject not recognised" let numStr := (toString num).leftpad 2 '0'; markdownTable := markdownTable.append s!"\| {count} \| {numStr} \|{desc} \|\n" return markdownTable -- TODO(firsching): instead of re-inventing the wheel here use some html parsing library?	def	docbuild	[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]	docbuild/scripts/overwrite_index.lean	getSubjectStatsMarkdown	null
replaceTag (tag : String) (inputHtmlContent : String) (newContent : String) : IO String := do let openTag := s!"<{tag}>" let closeTag := s!"</{tag}>" -- Find the position right after "<tag>" let .some bodyOpenTagSubstring := inputHtmlContent.findSubstr? openTag \| throw <\| IO.userError s!"Opening {openTag} tag not found in inputHtmlContent." let contentStartIndex := bodyOpenTagSubstring.stopPos -- Find the position of "</tag>" let .some bodyCloseTagSubstring := inputHtmlContent.findSubstr? closeTag \| throw <\| IO.userError s!"Closing {closeTag} tag not found in inputHtmlContent." -- Ensure the tags are in the correct order if contentStartIndex > bodyCloseTagSubstring.startPos then throw <\| IO.userError s!"{openTag} content appears invalid (start of content is after start of {closeTag} tag)." -- Extract the part of the HTML before the original body content (includes "<tag>") let htmlPrefix := inputHtmlContent.extract 0 contentStartIndex -- Extract the part of the HTML from "</tag>" to the end let htmlSuffix := inputHtmlContent.extract bodyCloseTagSubstring.startPos inputHtmlContent.endPos -- Construct the new full HTML content let finalHtml := htmlPrefix ++ newContent ++ htmlSuffix return finalHtml /-- Runs a `CoreM α` action in an environment where all FormalConjectures modules are imported. This is useful for accessing declarations and attributes defined in the project. -/	def	docbuild	[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]	docbuild/scripts/overwrite_index.lean	replaceTag	null
runWithImports {α : Type} (actionToRun : CoreM α) : IO α := do -- This assumes a run of `lake exe mk_all; mv FormalConjectures.lean FormalConjectures/All.lean` took place before. -- TODO(firsching): avoid this by instead using `Lake.Glob.forEachModuleIn` to generate a list of all modules instead. -- Then it would be easily possible to sort out the statements from the Util dir (in tests), -- which we probably don't want to count here. let moduleImportNames := #[`FormalConjectures.All] initSearchPath (← findSysroot) let imports : Array Import := moduleImportNames.map ({ module := · }) let currentCtx := { fileName := "", fileMap := default } Lean.enableInitializersExecution let env ← Lean.importModules imports {} (trustLevel := 1024) (loadExts := true) let (result, _newState) ← Core.CoreM.toIO actionToRun currentCtx { env := env } return result	def	docbuild	[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]	docbuild/scripts/overwrite_index.lean	runWithImports	/-- Runs a `CoreM α` action in an environment where all FormalConjectures modules are imported. This is useful for accessing declarations and attributes defined in the project. -/
main (args : List String) : IO Unit := do let .some (file : String) := args[0]? \| IO.println "Usage: stats <file> overwrites the contents of the `main` tag of a html `file` with a welcome page including stats." let inputHtmlContent ← IO.FS.readFile file let .some (graphFile : String) := args[1]? \| IO.println "Repository growth graph not supplied, generating docs without graph." let graphHtml ← IO.FS.readFile graphFile runWithImports do let categoryStats ← getCategoryStatsMarkdown let subjectStats ← getSubjectStatsMarkdown let markdownBody := s!"# Welcome to the Formal Conjectures Documentation! Check out the main [Formal Conjectures GitHub repository](https://github.com/google-deepmind/formal-conjectures) for more details. This page provides an overview of the problem categories and subject classifications used within the project. For a more detailed explanation of these categories and the AMS subject classifications, please refer to the [explanation of features in the project's README](https://github.com/google-deepmind/formal-conjectures?tab=readme-ov-file#some-features). --- ## Problem Category Statistics {categoryStats} (note the links above use GitHub search, and so require logging into GitHub) --- ## Subject Category Statistics {subjectStats} --- ## Repository growth " IO.println markdownBody let .some newBody := MD4Lean.renderHtml (parserFlags := MD4Lean.MD_FLAG_TABLES ) markdownBody \| throwError "Parsing failed" let finalHtml ← replaceTag "main" inputHtmlContent (newBody ++ graphHtml) IO.FS.writeFile file finalHtml	def	docbuild	[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]	docbuild/scripts/overwrite_index.lean	main	null
IsSumDistinctSet (A : Finset ℕ) (N : ℕ) : Prop := A ⊆ Finset.Icc 1 N ∧ (fun (⟨S, _⟩ : A.powerset) => S.sum id).Injective /-- If $A\subseteq\{1, ..., N\}$ with $\|A\| = n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $$ N \gg 2 ^ n. $$ -/ @[category research open, AMS 5 11]	abbrev	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	IsSumDistinctSet	/-- A finite set of naturals $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A\subseteq\{1, ..., N\}$ and the sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ -/
erdos_1 : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℕ) (_ : IsSumDistinctSet A N), N ≠ 0 → C * 2 ^ A.card < N := by sorry /-- The trivial lower bound is $N \gg 2^n / n$. -/ @[category undergraduate, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	erdos_1	/-- If $A\subseteq\{1, ..., N\}$ with $\|A\| = n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $$ N \gg 2 ^ n. $$ -/
erdos_1.variants.weaker : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℕ) (_ : IsSumDistinctSet A N), N ≠ 0 → C * 2 ^ A.card / A.card < N := by sorry /-- Erdős and Moser [Er56] proved $$ N \geq (\tfrac{1}{4} - o(1)) \frac{2^n}{\sqrt{n}}. $$ [Er56] Erdős, P., _Problems and results in additive number theory_. Colloque sur la Th\'{E}orie des Nombres, Bruxelles, 1955 (1956), 127-137. -/ @[category research solved, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	erdos_1.variants.weaker	/-- The trivial lower bound is $N \gg 2^n / n$. -/
erdos_1.variants.lb : ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)), ∀ (N : ℕ) (A : Finset ℕ) (h : IsSumDistinctSet A N), (1 / 4 - o A.card) * 2 ^ A.card / (A.card : ℝ).sqrt ≤ N := by sorry /-- A number of improvements of the constant $\frac{1}{4}$ have been given, with the current record $\sqrt{2 / \pi}$ first provied in unpublished work of Elkies and Gleason. -/ @[category research solved, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	erdos_1.variants.lb	/-- Erdős and Moser [Er56] proved $$ N \geq (\tfrac{1}{4} - o(1)) \frac{2^n}{\sqrt{n}}. $$ [Er56] Erdős, P., _Problems and results in additive number theory_. Colloque sur la Th\'{E}orie des Nombres, Bruxelles, 1955 (1956), 127-137. -/
erdos_1.variants.lb_strong : ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)), ∀ (N : ℕ) (A : Finset ℕ) (h : IsSumDistinctSet A N), (√(2 / π) - o A.card) * 2 ^ A.card / (A.card : ℝ).sqrt ≤ N := by sorry /-- A finite set of real numbers is said to be sum-distinct if all the subset sums differ by at least $1$. -/	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	erdos_1.variants.lb_strong	/-- A number of improvements of the constant $\frac{1}{4}$ have been given, with the current record $\sqrt{2 / \pi}$ first provied in unpublished work of Elkies and Gleason. -/
IsSumDistinctRealSet (A : Finset ℝ) (N : ℕ) : Prop := A.toSet ⊆ Set.Ioc 0 N ∧ A.powerset.toSet.Pairwise fun S₁ S₂ => 1 ≤ dist (S₁.sum id) (S₂.sum id) /-- A generalisation of the problem to sets $A \subseteq (0, N]$ of real numbers, such that the subset sums all differ by at least $1$ is proposed in [Er73] and [ErGr80]. [Er73] Erdős, P., _Problems and results on combinatorial number theory_. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. [ErGr80] Erdős, P. and Graham, R., _Old and new problems and results in combinatorial number theory_. Monographies de L'Enseignement Mathematique (1980). -/ @[category research open, AMS 5 11]	abbrev	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	IsSumDistinctRealSet	/-- A finite set of real numbers is said to be sum-distinct if all the subset sums differ by at least $1$. -/
erdos_1.variants.real : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℝ) (_ : IsSumDistinctRealSet A N), N ≠ 0 → C * 2 ^ A.card < N := by sorry /-- The minimal value of $N$ such that there exists a sum-distinct set with three elements is $4$. https://oeis.org/A276661 -/ @[category undergraduate, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	erdos_1.variants.real	null
erdos_1.variants.least_N_3 : IsLeast { N \| ∃ A, IsSumDistinctSet A N ∧ A.card = 3 } 4 := by refine ⟨⟨{1, 2, 4}, ?_⟩, ?_⟩ · simp refine ⟨by decide, ?_⟩ let P := Finset.powerset {1, 2, 4} have : Finset.univ.image (fun p : P ↦ ∑ x ∈ p, x) = {0, 1, 2, 4, 3, 5, 6, 7} := by refine Finset.ext_iff.mpr (fun n => ?_) simp [show P = {{}, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}} by decide] omega rw [Set.injective_iff_injOn_univ, ← Finset.coe_univ] have : (Finset.univ.image (fun p : P ↦ ∑ x ∈ p.1, x)).card = (Finset.univ (α := P)).card := by rw [this]; aesop exact Finset.injOn_of_card_image_eq this · simp [mem_lowerBounds] intro n S h h_inj hcard3 by_contra hn interval_cases n; aesop; aesop · have := Finset.card_le_card h aesop · absurd h_inj rw [(Finset.subset_iff_eq_of_card_le (Nat.le_of_eq (by rw [hcard3]; decide))).mp h] decide /-- The minimal value of $N$ such that there exists a sum-distinct set with five elements is $13$. https://oeis.org/A276661 -/ @[category research solved, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	erdos_1.variants.least_N_3	/-- The minimal value of $N$ such that there exists a sum-distinct set with three elements is $4$. https://oeis.org/A276661 -/
erdos_1.variants.least_N_5 : IsLeast { N \| ∃ A, IsSumDistinctSet A N ∧ A.card = 5 } 13 := by sorry /-- The minimal value of $N$ such that there exists a sum-distinct set with nine elements is $161$. https://oeis.org/A276661 -/ @[category research solved, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	erdos_1.variants.least_N_5	/-- The minimal value of $N$ such that there exists a sum-distinct set with five elements is $13$. https://oeis.org/A276661 -/
erdos_1.variants.least_N_9 : IsLeast { N \| ∃ A, IsSumDistinctSet A N ∧ A.card = 9 } 161 := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1.lean	erdos_1.variants.least_N_9	/-- The minimal value of $N$ such that there exists a sum-distinct set with nine elements is $161$. https://oeis.org/A276661 -/
sumPrimeAndTwoPows (k : ℕ) : Set ℕ := { p + (pows.map (2 ^ ·)).sum \| (p : ℕ) (pows : Multiset ℕ) (_ : p.Prime) (_ : pows.card ≤ k)} /-- Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$? -/ @[category research open, AMS 5 11]	abbrev	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/10.lean	sumPrimeAndTwoPows	/-- The set of natural numbers that can be written as a sum of a prime and at most $k$ powers of $2$. -/
erdos_10 : answer(sorry) ↔ ∃ k, sumPrimeAndTwoPows k = Set.univ \ {0, 1} := by sorry /-- Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$ such that the set of integers which are the sum of a prime and at most $k(ϵ)$ many powers of $2$ has lower density at least $1 - ϵ$. Ref: Gallagher, P. X., _Primes and powers of 2_. -/ @[category research solved, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/10.lean	erdos_10	/-- Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$? -/
erdos_10.variants.gallagher (ε : ℝ) (hε : 0 < ε) : ∃ k, 1 - ε ≤ (sumPrimeAndTwoPows k).lowerDensity := by sorry /-- Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. Ref: Granville, A. and Soundararajan, K., _A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$_ -/ @[category research open, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/10.lean	erdos_10.variants.gallagher	/-- Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$ such that the set of integers which are the sum of a prime and at most $k(ϵ)$ many powers of $2$ has lower density at least $1 - ϵ$. Ref: Gallagher, P. X., _Primes and powers of 2_. -/
erdos_10.variants.granville_soundararajan_odd : {n : ℕ \| Odd n ∧ 1 < n} ⊆ sumPrimeAndTwoPows 3 ∧ {n : ℕ \| Even n ∧ n ≠ 0} ⊆ sumPrimeAndTwoPows 4 := by sorry /-- Bogdan Grechuk has observed that `1117175146` is not the sum of a prime and at most $3$ powers of $2$. -/ @[category research solved, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/10.lean	erdos_10.variants.granville_soundararajan_odd	/-- Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. Ref: Granville, A. and Soundararajan, K., _A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$_ -/
erdos_10.variants.grechuk_example : 1117175146 ∉ sumPrimeAndTwoPows 3 := by sorry /-- There are infinitely many even integers not the sum of a prime and $2$ powers of $2$ -/ @[category research solved, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/10.lean	erdos_10.variants.grechuk_example	/-- Bogdan Grechuk has observed that `1117175146` is not the sum of a prime and at most $3$ powers of $2$. -/
erdos_10.variants.two_pows : Set.Infinite <\| {n : ℕ \| Even n} \ sumPrimeAndTwoPows 2 := by sorry /-- Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$). -/ @[category research open, AMS 5 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/10.lean	erdos_10.variants.two_pows	/-- There are infinitely many even integers not the sum of a prime and $2$ powers of $2$ -/
erdos_10.variants.gretchuk : Set.Infinite <\| {n : ℕ \| Even n} \ sumPrimeAndTwoPows 3 := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/10.lean	erdos_10.variants.gretchuk	/-- Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$). -/
erdos_1003 : answer(sorry) ↔ Set.Infinite {n \| φ n = φ (n + 1)} := by sorry /-- Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions. [Er85e] Erdős, P., _Some problems and results in number theory_. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. -/ @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1003.lean	erdos_1003	/-- Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function? -/
erdos_1003.variants.Icc : answer(sorry) ↔ ∀ k ≥ 1, {n \| ∀ i ∈ Set.Icc 1 k, φ n = φ (n + i)}.Infinite := by sorry /-- Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$. [EPS87] Erd\H os, Paul and Pomerance, Carl and S\'ark\"ozy, Andr\'as, _On locally repeated values of certain arithmetic functions_. {II}. Proc. Amer. Math. Soc. (1987), 1--7. -/ @[category research solved, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1003.lean	erdos_1003.variants.Icc	/-- Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions. [Er85e] Erdős, P., _Some problems and results in number theory_. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. -/
erdos_1003.variants.eps87 : ∀ᶠ x in atTop, {(n : ℕ) \| (n ≤ x) ∧ φ n = φ (n + 1)}.ncard ≤ x / Real.exp ((x.log) ^ ((1 : ℝ) / 3)) := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1003.lean	erdos_1003.variants.eps87	/-- Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$. [EPS87] Erd\H os, Paul and Pomerance, Carl and S\'ark\"ozy, Andr\'as, _On locally repeated values of certain arithmetic functions_. {II}. Proc. Amer. Math. Soc. (1987), 1--7. -/
IsDistinctTotientRun (n K : ℕ) : Prop := (Set.Icc (n + 1) (n + K)).InjOn totient /-- For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem. -/ @[category research open, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1004.lean	IsDistinctTotientRun	/-- `IsDistinctTotientRun n K` means that the values `φ(n+1), φ(n+2), ..., φ(n+K)` are all distinct. -/
erdos_1004 : answer(sorry) ↔ ∀ c > (0 : ℝ), ∀ᶠ x in atTop, ∃ n ≤ x, IsDistinctTotientRun n ⌊(Real.log (x : ℝ)) ^ c⌋₊ := by sorry /-- Erdős, Pomerance, and Sárközy [EPS87] proved that if φ(n+k) are all distinct for 1 ≤ k ≤ K then K ≤ n / exp(c (log n)^{1/3}) for some constant c > 0. Here we state the existence of such a constant c. -/ @[category research solved, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1004.lean	erdos_1004	/-- For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem. -/
erdos_1004.EPS87_theorem : answer(True) ↔ ∃ (c : ℝ) (hc : c > 0), ∀ (n K : ℕ), n > 0 → IsDistinctTotientRun n K → (K : ℝ) ≤ (n : ℝ) / Real.exp (c * (Real.log n) ^ (1/3 : ℝ)) := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1004.lean	erdos_1004.EPS87_theorem	/-- Erdős, Pomerance, and Sárközy [EPS87] proved that if φ(n+k) are all distinct for 1 ≤ k ≤ K then K ≤ n / exp(c (log n)^{1/3}) for some constant c > 0. Here we state the existence of such a constant c. -/
erdos_1038.inf (n : ℕ) : answer(sorry) = ⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧ (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree}, volume {x \| \|f.1.eval x\| < 1} := by sorry /-- The infimum of `\|{x ∈ ℝ : \|f x\| < 1}\|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `< 1.835`. -/ @[category research solved, AMS 28]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree" ]	FormalConjectures/ErdosProblems/1038.lean	erdos_1038.inf	/-- What is the infimum of `\|{x ∈ ℝ : \|f x\| < 1}\|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]`? -/
erdos_1038.inf_upperBound (n : ℕ) : ⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧ (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree}, volume {x \| \|f.1.eval x\| < 1} < 1.835 := by sorry /-- The infimum of `\|{x ∈ ℝ : \|f x\| < 1}\|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `≥ 2 ^ (4 / 3) - 1`. -/ @[category research solved, AMS 28]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree" ]	FormalConjectures/ErdosProblems/1038.lean	erdos_1038.inf_upperBound	/-- The infimum of `\|{x ∈ ℝ : \|f x\| < 1}\|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `< 1.835`. -/
erdos_1038.inf_lowerBound (n : ℕ) : 2 ^ (4 / 3 : ℝ) - 1 ≤ ⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧ (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree}, volume {x \| \|f.1.eval x\| < 1} := by sorry /-- The supremum of `\|{x ∈ ℝ : \|f x\| < 1}\|` over all monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `2 * 2 ^ (1 / 2)`. This is proved in [Tao25]. -/ @[category research solved, AMS 28]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree" ]	FormalConjectures/ErdosProblems/1038.lean	erdos_1038.inf_lowerBound	/-- The infimum of `\|{x ∈ ℝ : \|f x\| < 1}\|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `≥ 2 ^ (4 / 3) - 1`. -/
erdos_1038.sup (n : ℕ) : 2 * 2 ^ (1 / 2 : ℝ) = ⨆ f : {f : Polynomial ℝ // f.Monic ∧ (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree}, volume {x \| \|f.1.eval x\| < 1} := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree" ]	FormalConjectures/ErdosProblems/1038.lean	erdos_1038.sup	/-- The supremum of `\|{x ∈ ℝ : \|f x\| < 1}\|` over all monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `2 * 2 ^ (1 / 2)`. This is proved in [Tao25]. -/
length (s : Set ℂ) : ℝ≥0∞ := μH[1] s /-- Erdős–Herzog–Piranian Component Lemma (Metric Properties of Polynomials, 1958): If $f$ is a monic degree $n$ polynomial with all roots in the unit disk, then some connected component of $\{z \mid \|f(z)\| < 1\}$ contains at least two roots with multiplicity. See p. 139, above Problem 5: [EHP58] Erdős, P. and Herzog, F. and Piranian, G., _Metric properties of polynomials_. J. Analyse Math. (1958), 125-148. -/ @[category research solved, AMS 32]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1041.lean	length	/-- The length of a subset $s$ of $\mathbb{C}$ is defined to be its 1-dimensional Hausdorff measure $\mathcal{H}^1(s)$. -/
exists_connected_component_contains_two_roots : ∃ C, C ⊆ {z \| ‖f.eval z‖ < 1} ∧ IsConnected C ∧ 2 ≤ (f.roots.filter (· ∈ C)).card := by sorry /-- Let $$ f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x] $$ with $\|z_i\| < 1$ for all $i$. Conjecture: Must there always exist a path of length less than 2 in $$ \{ z \in \mathbb{C} \mid \|f(z)\| < 1 \} $$ which connects two of the roots of $f$? -/ @[category research open, AMS 32]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1041.lean	exists_connected_component_contains_two_roots	/-- Erdős–Herzog–Piranian Component Lemma (Metric Properties of Polynomials, 1958): If $f$ is a monic degree $n$ polynomial with all roots in the unit disk, then some connected component of $\{z \mid \|f(z)\| < 1\}$ contains at least two roots with multiplicity. See p. 139, above Problem 5: [EHP58] Erdős, P. and Herzog, F. and Piranian, G., _Metric properties of polynomials_. J. Analyse Math. (1958), 125-148. -/
erdos_1041 : ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂), Set.range γ ⊆ { z : ℂ \| ‖f.eval z‖ < 1 } ∧ length (Set.range γ) < 2 := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1041.lean	erdos_1041	/-- Let $$ f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x] $$ with $\|z_i\| < 1$ for all $i$. Conjecture: Must there always exist a path of length less than 2 in $$ \{ z \in \mathbb{C} \mid \|f(z)\| < 1 \} $$ which connects two of the roots of $f$? -/
levelSet (f : Polynomial ℂ) : Set ℂ := {z : ℂ \| ‖f.eval z‖ ≤ 1} /-- Erdős Problem 1043: Let $f\in \mathbb{C}[x]$ be a monic polynomial. Must there exist a straight line $\ell$ such that the projection of \[\{ z: \lvert f(z)\rvert\leq 1\}\] onto $\ell$ has measure at most $2$? Pommerenke [Po61] proved that the answer is no. [Po61] Pommerenke, Ch., _On metric properties of complex polynomials._ Michigan Math. J. (1961), 97-115. -/ @[category research solved, AMS 28 30]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1043.lean	levelSet	/-- The set $\{ z \in \mathbb{C} : \lvert f(z)\rvert\leq 1\}$ -/
erdos_1043 : answer(False) ↔ ∀ (f : ℂ[X]), f.Monic → f.degree ≥ 1 → ∃ (u : ℂ), ‖u‖ = 1 ∧ volume ((ℝ ∙ u).orthogonalProjection '' levelSet f) ≤ 2 := by sorry /-- On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most 3.3. -/ @[category research solved, AMS 28 30]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1043.lean	erdos_1043	/-- Erdős Problem 1043: Let $f\in \mathbb{C}[x]$ be a monic polynomial. Must there exist a straight line $\ell$ such that the projection of \[\{ z: \lvert f(z)\rvert\leq 1\}\] onto $\ell$ has measure at most $2$? Pommerenke [Po61] proved that the answer is no. [Po61] Pommerenke, Ch., _On metric properties of complex polynomials._ Michigan Math. J. (1961), 97-115. -/
erdos_1043.variants.weak : ∀ (f : ℂ[X]), f.Monic → f.degree ≥ 1 → ∃ (u : ℂ), ‖u‖ = 1 ∧ volume ((ℝ ∙ u).orthogonalProjection '' levelSet f) ≤ 3.3 := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1043.lean	erdos_1043.variants.weak	/-- On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most 3.3. -/
GrowthCondition (a : ℕ → ℤ) : Prop := Filter.liminf (fun n => ((a n : ℝ) ^ (1 / 2 ^ n : ℝ))) Filter.atTop > 1 /-- The series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$. -/	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1051.lean	GrowthCondition	/-- A sequence of integers `a` satisfies the growth condition if $\liminf a_n^{\frac{1}{2^n}} > 1$. -/
ErdosSeries (a : ℕ → ℤ) : ℝ := ∑' n : ℕ, 1 / ((a n : ℝ) * (a (n + 1) : ℝ)) /-- Is it true that if $a_0 < a_1 < a_2 < \cdots$ is a strictly increasing sequence of integers with $\liminf a_n^{1/2^n} > 1$, then the series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$ is irrational? -/ @[category research open, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1051.lean	ErdosSeries	/-- The series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$. -/
erdos_1051 : answer(sorry) ↔ ∀ (a : ℕ → ℤ), StrictMono a → GrowthCondition a → Irrational (ErdosSeries a) := by sorry /-- Erdős [Er88c] notes that if the sequence grows rapidly to infinity (specifically, if $a_{n+1} \geq C \cdot a_n^2$ for some constant $C > 0$), then the series is irrational. [Er88c] Erdős, P., _On the irrationality of certain series: problems and results_. New advances in transcendence theory (Durham, 1986) (1988), 102-109. -/ @[category research solved, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1051.lean	erdos_1051	/-- Is it true that if $a_0 < a_1 < a_2 < \cdots$ is a strictly increasing sequence of integers with $\liminf a_n^{1/2^n} > 1$, then the series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$ is irrational? -/
erdos_1051.rapid_growth (a : ℕ → ℤ) (h_mono : StrictMono a) (h_rapid : ∃ C > 0, ∀ n, (a (n + 1) : ℝ) ≥ C * (a n : ℝ) ^ 2) : Irrational (ErdosSeries a) := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1051.lean	erdos_1051.rapid_growth	/-- Erdős [Er88c] notes that if the sequence grows rapidly to infinity (specifically, if $a_{n+1} \geq C \cdot a_n^2$ for some constant $C > 0$), then the series is irrational. [Er88c] Erdős, P., _On the irrationality of certain series: problems and results_. New advances in transcendence theory (Durham, 1986) (1988), 102-109. -/
properUnitaryDivisors (n : ℕ) : Finset ℕ := {d ∈ Finset.Ico 1 n \| d ∣ n ∧ d.Coprime (n / d)} /-- A number $n > 0$ is a unitary perfect number if it is the sum of its proper unitary divisors. -/	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	properUnitaryDivisors	/-- A proper unitary divisor of $n$ is a divisor $d$ of $n$ such that $d$ is coprime to $n/d$, and $d < n$. -/
IsUnitaryPerfect (n : ℕ) : Prop := ∑ i ∈ properUnitaryDivisors n, i = n ∧ 0 < n /-- Are there only finitely many unitary perfect numbers? -/ @[category research open, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	IsUnitaryPerfect	/-- A number $n > 0$ is a unitary perfect number if it is the sum of its proper unitary divisors. -/
erdos_1052 : answer(sorry) ↔ {n \| IsUnitaryPerfect n}.Finite := by sorry /-- All unitary perfect numbers are even. -/ @[category research solved, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	erdos_1052	/-- Are there only finitely many unitary perfect numbers? -/
even_of_isUnitaryPerfect (n : ℕ) (hn : IsUnitaryPerfect n) : Even n := by sorry @[category test, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	even_of_isUnitaryPerfect	/-- All unitary perfect numbers are even. -/
isUnitaryPerfect_6 : IsUnitaryPerfect 6 := by norm_num [IsUnitaryPerfect, properUnitaryDivisors] decide +kernel @[category test, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	isUnitaryPerfect_6	/-- All unitary perfect numbers are even. -/
isUnitaryPerfect_60 : IsUnitaryPerfect 60 := by norm_num [IsUnitaryPerfect, properUnitaryDivisors] decide +kernel @[category test, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	isUnitaryPerfect_60	/-- All unitary perfect numbers are even. -/
isUnitaryPerfect_90 : IsUnitaryPerfect 90 := by norm_num [IsUnitaryPerfect, properUnitaryDivisors] decide +kernel @[category test, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	isUnitaryPerfect_90	/-- All unitary perfect numbers are even. -/
isUnitaryPerfect_87360 : IsUnitaryPerfect 87360 := by norm_num [IsUnitaryPerfect, properUnitaryDivisors] decide +kernel @[category test, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	isUnitaryPerfect_87360	null
isUnitaryPerfect_146361946186458562560000 : IsUnitaryPerfect 146361946186458562560000 := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1052.lean	isUnitaryPerfect_146361946186458562560000	null
f (n : ℕ) : ℕ := if h : ∃ᵉ (m) (k ≥ 1), n = ∑ i < k, Nat.nth (· ∈ m.divisors) i then Nat.find h else 0 /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?-/ @[category research open, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1054.lean	f	/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$.-/
erdos_1054.parts.i : answer(sorry) ↔ (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := by sorry /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$? -/ @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1054.lean	erdos_1054.parts.i	/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?-/
erdos_1054.parts.ii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧ (fun (n : A) ↦ (f ↑n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := by sorry /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$? -/ @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1054.lean	erdos_1054.parts.ii	/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$? -/
erdos_1054.parts.iii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧ atTop.limsup (fun n ↦ (f n : EReal) / n) = ⊤ := by sorry /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$. -/ @[category high_school, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1054.lean	erdos_1054.parts.iii	/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$? -/
f_undefined_at_2 : f 2 = 0 := by sorry /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$. -/ @[category high_school, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1054.lean	f_undefined_at_2	/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$. -/
f_undefined_at_3 : f 5 = 0 := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1054.lean	f_undefined_at_3	/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$. -/
IsOfClass : ℕ+ → ℕ → Prop := fun r ↦ PNat.caseStrongInductionOn (p := fun (_ : ℕ+) ↦ ℕ → Prop) r (fun p ↦ (p + 1).primeFactors ⊆ {2, 3}) (fun n H p ↦ (∀ r ∈ (p + 1).primeFactors, ∃ (m : ℕ+) (hm : m ≤ n), H m hm r) ∧ (∃ r ∈ (p + 1).primeFactors, ∀ (m : ℕ+) (hm : m ≤ n), H m hm r → m = n)) /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Show that for each $r$ there exists a prime $p$ of class $r$. -/ @[category undergraduate, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1055.lean	IsOfClass	/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor.-/
exists_p (r : ℕ+) : ∃ p, p.Prime ∧ IsOfClass r p := by sorry open Classical /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Let $p_r$ is the least prime in class $r$.-/	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1055.lean	exists_p	/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Show that for each $r$ there exists a prime $p$ of class $r$. -/
p (r : ℕ+) : ℕ := Nat.find (exists_p r) /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Are there infinitely many primes in each class?-/ @[category research open, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1055.lean	p	/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Let $p_r$ is the least prime in class $r$.-/
erdos_1055 (r) : {p \| p.Prime ∧ IsOfClass r p}.Infinite := by sorry /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Erdos conjectured that this tends to infinity. -/ @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1055.lean	erdos_1055	/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Are there infinitely many primes in each class?-/
erdos_1055.variants.erdos_limit : Filter.atTop.Tendsto (fun r ↦ (p r : ℝ) ^ (1 / r : ℝ)) Filter.atTop := by sorry /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Selfridge conjectured that this is bounded. -/ @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1055.lean	erdos_1055.variants.erdos_limit	/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Erdos conjectured that this tends to infinity. -/
erdos_1055.variants.selfridge_limit : ∃ M, ∀ r, (p r : ℝ) ^ (1 / r : ℝ) ≤ M := by sorry --TODO(Paul-Lez): formalize the rest of the problems on the page.	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1055.lean	erdos_1055.variants.selfridge_limit	/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Selfridge conjectured that this is bounded. -/
AllModProdEqualsOne (p : ℕ) {k : ℕ} (boundaries : Fin (k + 1) → ℕ) : Prop := ∀ i : Fin k, (∏ n ∈ Finset.Ico (boundaries i.castSucc) (boundaries (i.castSucc + 1)), n) ≡ 1 [MOD p] /-- Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$ such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$? -/ @[category research open, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1056.lean	AllModProdEqualsOne	/-- The proposition that the modular product of a collection of consecutive interval equals $1$ modulo $p$, where intervals are defined by a function specifying the consecutive boundaries. -/
erdos_1056 : answer(sorry) ↔ ∀ k ≥ 2, ∃ (p : ℕ) (_ : p.Prime) (boundaries : Fin (k + 1) → ℕ) (_ : StrictMono boundaries), AllModProdEqualsOne p boundaries := by sorry /-- This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979 Erdős observed that $3 * 4 \equiv 5 * 6 * 7 \equiv 1 \mod 11$. -/ @[category undergraduate, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1056.lean	erdos_1056	/-- Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$ such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$? -/
erdos_1056_k2 : AllModProdEqualsOne 11 ![3, 5, 8] := by unfold AllModProdEqualsOne decide /-- Makowski [Ma83] found, for $k=3$: $2 * 3 * 4 * 5 \equiv 6 * 7 * 8 * 9 * 10 * 11 \equiv 12 * 13 * 14 * 15 \equiv 1 \mod 17$. -/ @[category undergraduate, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1056.lean	erdos_1056_k2	/-- This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979 Erdős observed that $3 * 4 \equiv 5 * 6 * 7 \equiv 1 \mod 11$. -/
erdos_1056_k3 : AllModProdEqualsOne 17 ![2, 6, 12, 16] := by unfold AllModProdEqualsOne decide /-- Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$). -/ @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1056.lean	erdos_1056_k3	/-- Makowski [Ma83] found, for $k=3$: $2 * 3 * 4 * 5 \equiv 6 * 7 * 8 * 9 * 10 * 11 \equiv 12 * 13 * 14 * 15 \equiv 1 \mod 17$. -/
noll_simmons : answer(sorry) ↔ ∀ᶠ k in Filter.atTop, ∃ (p : ℕ) (_ : p.Prime) (Q : Fin k → ℕ) (_ : StrictMono Q) (_ : ∀ i, Q i < p), ∀ i j : Fin k, (Q i)! ≡ (Q j)! [MOD p] := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1056.lean	noll_simmons	/-- Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$). -/
IsFactorial (d : ℕ) : Prop := d ∈ Set.range Nat.factorial	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	IsFactorial	null
factorialsLessThanN (n : ℕ) : Set ℕ := { d \| d < n ∧ IsFactorial d }	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	factorialsLessThanN	null
AllFactorialSubtractionsComposite (n : ℕ) : Prop := ∀d ∈ factorialsLessThanN n, (n - d).Composite /-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/ @[category research open, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	AllFactorialSubtractionsComposite	null
erdos_1059 : answer(sorry) ↔ Set.Infinite {p \| p.Prime ∧ AllFactorialSubtractionsComposite p} := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	erdos_1059	/-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/
DecidableIsFactorial (d : ℕ) : Prop := ((Finset.Icc 0 d).filter (λ k => Nat.factorial k = d)).Nonempty	abbrev	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	DecidableIsFactorial	/-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/
decidableFactorialsLessThanN (n : ℕ) : Finset ℕ := (Finset.range n).filter DecidableIsFactorial	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	decidableFactorialsLessThanN	/-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/
DecidableAllFactorialSubtractionsComposite (n : ℕ) : Prop := ∀ d ∈ decidableFactorialsLessThanN n, (n - d).Composite @[category test, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	DecidableAllFactorialSubtractionsComposite	/-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/
isFactorial_equivalent (d : ℕ) : IsFactorial d ↔ DecidableIsFactorial d := by unfold IsFactorial DecidableIsFactorial simp constructor · rintro ⟨k, hk⟩ use k rw [Finset.mem_filter] constructor · have hk : k <= d := by rw [← hk] apply Nat.self_le_factorial rw [Finset.mem_Icc] exact ⟨Nat.zero_le k, hk⟩ · exact hk · rintro ⟨k, hk⟩ use k rw [Finset.mem_filter] at hk exact hk.2 @[category test, AMS 11]	lemma	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	isFactorial_equivalent	null
factorialsLessThanN_equivalent (n : ℕ) : factorialsLessThanN n = ↑(decidableFactorialsLessThanN n) := by ext d unfold factorialsLessThanN decidableFactorialsLessThanN simp exact λ _ => isFactorial_equivalent d @[category test, AMS 11]	lemma	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	factorialsLessThanN_equivalent	null
allFactorialSubtractionsComposite_equivalent (d : ℕ) : DecidableAllFactorialSubtractionsComposite d ↔ AllFactorialSubtractionsComposite d := by unfold AllFactorialSubtractionsComposite DecidableAllFactorialSubtractionsComposite rw [factorialsLessThanN_equivalent d] simp @[category test, AMS 11]	lemma	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	allFactorialSubtractionsComposite_equivalent	null
allFactorialSubtractionsComposite_101 : AllFactorialSubtractionsComposite 101 := by have h : DecidableAllFactorialSubtractionsComposite 101 := by norm_num [DecidableAllFactorialSubtractionsComposite, decidableFactorialsLessThanN] decide +kernel exact (allFactorialSubtractionsComposite_equivalent 101).mp h @[category test, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	allFactorialSubtractionsComposite_101	null
allFactorialSubtractionsComposite_211 : AllFactorialSubtractionsComposite 211 := by have h : DecidableAllFactorialSubtractionsComposite 211 := by norm_num [DecidableAllFactorialSubtractionsComposite, decidableFactorialsLessThanN] decide +kernel exact (allFactorialSubtractionsComposite_equivalent 211).mp h @[category test, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	allFactorialSubtractionsComposite_211	null
notAllFactorialSubtractionsComposite_89 : ¬(AllFactorialSubtractionsComposite 89) := by have h : ¬(DecidableAllFactorialSubtractionsComposite 89) := by unfold DecidableAllFactorialSubtractionsComposite decidableFactorialsLessThanN intro h specialize h 6 have : Nat.Prime (89 - 6) := by norm_num contradiction simp [allFactorialSubtractionsComposite_equivalent] at h exact h @[category test, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	notAllFactorialSubtractionsComposite_89	null
testFactorialsLessThanN : factorialsLessThanN 100 = {1, 2, 6, 24} := by have h : decidableFactorialsLessThanN 100 = {1, 2, 6, 24} := by norm_num [decidableFactorialsLessThanN] decide +kernel rw [factorialsLessThanN_equivalent] simp [h]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1059.lean	testFactorialsLessThanN	null
erdos_1060.bound_one : ∃ h : ℕ → ℝ, h =o[atTop] (fun n ↦ 1 / log (log n)) ∧ ∀ᶠ n in atTop, #{k ≤ n \| k * σ 1 k = n} ≤ (n : ℝ) ^ h n := by sorry @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1060.lean	erdos_1060.bound_one	/-- The conjecture is about the function $f(n)$ which counts the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower than any power of $n^(\frac{1}{\log\log n})$. The second bound is that $f(n)$ is at most a power of $\log n$. -/
erdos_1060.bound_two : ∃ (C : ℝ), (fun n ↦ (#{k ≤ n \| k * σ 1 k = n} : ℝ)) =O[atTop] (fun n ↦ log n ^ C) := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1060.lean	erdos_1060.bound_two	null
ForkFree (A : Set ℕ) : Prop := ∀ a ∈ A, ({b \| b ∈ A \ {a} ∧ a ∣ b} : Set ℕ).Subsingleton open scoped Classical in /-- The extremal function from Erdős problem 1062: the largest size of a fork-free subset of `{1,...,n}`. -/	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]	FormalConjectures/ErdosProblems/1062.lean	ForkFree	/-- A set `A` of positive integers is fork-free if no element divides two distinct other elements of `A`. -/
f (n : ℕ) : ℕ := Nat.findGreatest (fun k => ∃ A ⊆ Set.Icc 1 n, ForkFree A ∧ A.ncard = k) n /-- The interval `[⌊n/3⌋, n]` is fork-free, and therefore `f n` is at least `⌈2n / 3⌉`. -/ @[category research solved, AMS 11]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]	FormalConjectures/ErdosProblems/1062.lean	f	/-- The extremal function from Erdős problem 1062: the largest size of a fork-free subset of `{1,...,n}`. -/
erdos_1062.lower_bound (n : ℕ) : ⌈(2 * n / 3 : ℝ)⌉₊ ≤ f n := by classical set b : ℕ := n / 3 with hb let A : Finset ℕ := .Icc (b + 1) n calc ⌈(2 * n / 3 : ℝ)⌉₊ ≤ n - b := by grw [Nat.ceil_le, Nat.cast_sub (by omega), le_sub_iff_add_le, hb, Nat.cast_div_le] -- FIXME: `ring` should have some basic inequality support. apply le_of_eq ring _ ≤ f n := Nat.le_findGreatest (by omega) ⟨A, by simp only [Finset.coe_Icc, A]; gcongr; omega, ?_, by simp [A, -Finset.coe_Icc]⟩ simp only [ForkFree, Finset.coe_Icc, Set.mem_Icc, Set.mem_diff, Set.mem_singleton_iff, and_assoc, and_imp, A] rintro a ha - refine Set.subsingleton_of_forall_eq (a * 2) ?_ simp only [Set.mem_setOf_eq, and_imp] rintro _ _ hk _ ⟨k, rfl⟩ match k with \| 0 \| 1 \| 2 => simp_all \| k + 3 => grw [← le_add_self] at hk; omega /-- Lebensold proved that for large `n`, the function `f n` lies between `0.6725 n` and `0.6736 n`. -/ @[category research solved, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]	FormalConjectures/ErdosProblems/1062.lean	erdos_1062.lower_bound	/-- The interval `[⌊n/3⌋, n]` is fork-free, and therefore `f n` is at least `⌈2n / 3⌉`. -/
erdos_1062.lebensold_bounds : ∀ᶠ n in atTop, (0.6725 : ℝ) * n ≤ f n ∧ f n ≤ (0.6736 : ℝ) * n := by sorry /-- Erdős asked whether the limiting density `f n / n` exists and, if so, whether it is irrational. -/ @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]	FormalConjectures/ErdosProblems/1062.lean	erdos_1062.lebensold_bounds	/-- Lebensold proved that for large `n`, the function `f n` lies between `0.6725 n` and `0.6736 n`. -/
erdos_1062.limit_density : (∃ l, Tendsto (fun n => (f n : ℝ) / n) atTop (𝓝 l) ∧ Irrational l) ↔ answer(sorry) := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]	FormalConjectures/ErdosProblems/1062.lean	erdos_1062.limit_density	/-- Erdős asked whether the limiting density `f n / n` exists and, if so, whether it is irrational. -/
erdos_1064 : {n \| φ n > φ (n - φ n)}.HasDensity 1 := by sorry /-- Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$ such that $ϕ(n)< ϕ(n - ϕ(n))$ Reference: [GLW01] Grytczuk, A. and Luca, F. and W\'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the {E}uler totient function. -/ @[category research solved, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1064.lean	erdos_1064	/-- Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$ have asymptotic density 1. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. -/
erdos_1064.variants.k2 : {n \| φ n < φ (n - φ n)}.Infinite := by sorry open Asymptotics Filter /-- For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. (2002), 111--130. -/ @[category research solved, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1064.lean	erdos_1064.variants.k2	/-- Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$ such that $ϕ(n)< ϕ(n - ϕ(n))$ Reference: [GLW01] Grytczuk, A. and Luca, F. and W\'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the {E}uler totient function. -/
erdos_1064.variants.general_function (f : ℕ → ℕ) (hf : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))) : {n : ℕ \| φ (n - φ n) + f n < φ n}.HasDensity 1 := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1064.lean	erdos_1064.variants.general_function	/-- For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. (2002), 111--130. -/
erdos_1065a : answer(sorry) ↔ Set.Infinite {p \| ∃ q k, p.Prime ∧ q.Prime ∧ p = 2^k * q + 1} := by sorry /-- Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$ for some prime $q$ and $k ≥ 0$, $l ≥ 0$? -/ @[category research open, AMS 11]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1065.lean	erdos_1065a	/-- Are there infinitely many primes $p$ such that $p = 2^k * q + 1$ for some prime $q$ and $k ≥ 0$? This is mentioned as B46 in [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0) by Richard K. Guy -/
erdos_1065b : answer(sorry) ↔ Set.Infinite {p \| ∃ q k l, p.Prime ∧ q.Prime ∧ p = 2^k * 3^l * q + 1} := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1065.lean	erdos_1065b	/-- Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$ for some prime $q$ and $k ≥ 0$, $l ≥ 0$? -/
InternallyDisjoint {V : Type*} {G : SimpleGraph V} {u v x y : V} (p : G.Walk u v) (q : G.Walk x y) : Prop := Disjoint p.support.tail.dropLast q.support.tail.dropLast /-- We say a graph is infinitely connected if any two vertices are connected by infinitely many pairwise disjoint paths. -/	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1068.lean	InternallyDisjoint	/-- Two walks are internally disjoint if they share no vertices other than their endpoints. -/
InfinitelyConnected {V : Type*} (G : SimpleGraph V) : Prop := Pairwise fun u v ↦ ∃ P : Set (G.Walk u v), P.Infinite ∧ (∀ p ∈ P, p.IsPath) ∧ P.Pairwise InternallyDisjoint /-- Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected? -/ @[category research open, AMS 5]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1068.lean	InfinitelyConnected	/-- We say a graph is infinitely connected if any two vertices are connected by infinitely many pairwise disjoint paths. -/
erdos_1068 : answer(sorry) ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticNumber = aleph 1 → ∃ s : Set V, s.Countable ∧ InfinitelyConnected (G.induce s) := by sorry	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/1068.lean	erdos_1068	/-- Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected? -/
cardSet (n : ℕ) := { N \| ∀ (pts : Finset ℝ²), pts.card = N → NonTrilinear pts.toSet → HasConvexNGon n pts } /-- The function $f(n)$ specified in `erdos_107`. -/	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/107.lean	cardSet	/-- The set of $N$ such that any $N$ points in the plane, no three on a line, contain a convex $n$-gon. -/
f (n : ℕ) : ℕ := sInf (cardSet n) /-- Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n) = 2^{n-2} + 1$. -/ @[category research open, AMS 52]	def	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/107.lean	f	/-- The function $f(n)$ specified in `erdos_107`. -/
erdos_107 : answer(sorry) ↔ ∀ n ≥ 3, f n = 2^(n - 2) + 1 := by sorry /-- For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line, contain a convex $n$-gon. -/ @[category research solved, AMS 52]	theorem	FormalConjectures	[ "import FormalConjectures.Util.ProblemImports" ]	FormalConjectures/ErdosProblems/107.lean	erdos_107	/-- Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n) = 2^{n-2} + 1$. -/

getCategoryStatsMarkdown : CoreM String := do let stats ← getCategoryStats let githubSearchBaseUrl := "https://github.com/search?type=code&q=repo%3Agoogle-deepmind%2Fformal-conjectures+" return s!"| Count | Category | | ----- | ----------------- | | {stats (Category.research ProblemStatus.open)} | [Research (open)]({githubSearchBaseUrl}%22category+research+open%22)| | {stats (Category.research ProblemStatus.solved)} | [Research (solved)]({githubSearchBaseUrl}%22category+research+solved%22)| | {stats (Category.graduate)} | [Graduate]({githubSearchBaseUrl}%22category+graduate%22)| | {stats (Category.undergraduate)} | [Undergraduate]({githubSearchBaseUrl}%22category+undergraduate%22)| | {stats (Category.highSchool)} | [High School]({githubSearchBaseUrl}%22category+high_school%22)| | {stats (Category.API)} | [API]({githubSearchBaseUrl}%22category+API%22)| | {stats (Category.test)} | [Tests]({githubSearchBaseUrl}%22category+tests%22)|" -- TODO(firsching): make it possible to search for subjects in doc-gen4, likely depends on -- https://github.com/google-deepmind/formal-conjectures/issues/5

def

docbuild

[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]

docbuild/scripts/overwrite_index.lean

getCategoryStatsMarkdown

null

getSubjectStatsMarkdown : CoreM String := do let tags ← getSubjectTags let mut counts : Std.HashMap AMS Nat := {} for tag in tags do for subject in tag.subjects do counts := counts.insert subject (counts.getD subject 0 + 1) let sortedCounts := counts.toArray.qsort (lt := fun (_, c1) (_, c2) => c2 < c1) let mut markdownTable := "| Count | AMS # | Subject |\n" ++ "| ----- | ----- | ------- |\n " for (subject, count) in sortedCounts do if count > 0 then let desc ← subject.getDesc let some num := subject.toNat? | throwError "subject not recognised" let numStr := (toString num).leftpad 2 '0'; markdownTable := markdownTable.append s!"| {count} | {numStr} |{desc} |\n" return markdownTable -- TODO(firsching): instead of re-inventing the wheel here use some html parsing library?

def

docbuild

[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]

docbuild/scripts/overwrite_index.lean

getSubjectStatsMarkdown

null

replaceTag (tag : String) (inputHtmlContent : String) (newContent : String) : IO String := do let openTag := s!"<{tag}>" let closeTag := s!"</{tag}>" -- Find the position right after "<tag>" let .some bodyOpenTagSubstring := inputHtmlContent.findSubstr? openTag | throw <| IO.userError s!"Opening {openTag} tag not found in inputHtmlContent." let contentStartIndex := bodyOpenTagSubstring.stopPos -- Find the position of "</tag>" let .some bodyCloseTagSubstring := inputHtmlContent.findSubstr? closeTag | throw <| IO.userError s!"Closing {closeTag} tag not found in inputHtmlContent." -- Ensure the tags are in the correct order if contentStartIndex > bodyCloseTagSubstring.startPos then throw <| IO.userError s!"{openTag} content appears invalid (start of content is after start of {closeTag} tag)." -- Extract the part of the HTML before the original body content (includes "<tag>") let htmlPrefix := inputHtmlContent.extract 0 contentStartIndex -- Extract the part of the HTML from "</tag>" to the end let htmlSuffix := inputHtmlContent.extract bodyCloseTagSubstring.startPos inputHtmlContent.endPos -- Construct the new full HTML content let finalHtml := htmlPrefix ++ newContent ++ htmlSuffix return finalHtml /-- Runs a `CoreM α` action in an environment where all FormalConjectures modules are imported. This is useful for accessing declarations and attributes defined in the project. -/

def

docbuild

[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]

docbuild/scripts/overwrite_index.lean

replaceTag

null

runWithImports {α : Type} (actionToRun : CoreM α) : IO α := do -- This assumes a run of `lake exe mk_all; mv FormalConjectures.lean FormalConjectures/All.lean` took place before. -- TODO(firsching): avoid this by instead using `Lake.Glob.forEachModuleIn` to generate a list of all modules instead. -- Then it would be easily possible to sort out the statements from the Util dir (in tests), -- which we probably don't want to count here. let moduleImportNames := #[`FormalConjectures.All] initSearchPath (← findSysroot) let imports : Array Import := moduleImportNames.map ({ module := · }) let currentCtx := { fileName := "", fileMap := default } Lean.enableInitializersExecution let env ← Lean.importModules imports {} (trustLevel := 1024) (loadExts := true) let (result, _newState) ← Core.CoreM.toIO actionToRun currentCtx { env := env } return result

def

docbuild

[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]

docbuild/scripts/overwrite_index.lean

runWithImports

/-- Runs a `CoreM α` action in an environment where all FormalConjectures modules are imported. This is useful for accessing declarations and attributes defined in the project. -/

main (args : List String) : IO Unit := do let .some (file : String) := args[0]? | IO.println "Usage: stats <file> overwrites the contents of the `main` tag of a html `file` with a welcome page including stats." let inputHtmlContent ← IO.FS.readFile file let .some (graphFile : String) := args[1]? | IO.println "Repository growth graph not supplied, generating docs without graph." let graphHtml ← IO.FS.readFile graphFile runWithImports do let categoryStats ← getCategoryStatsMarkdown let subjectStats ← getSubjectStatsMarkdown let markdownBody := s!"# Welcome to the *Formal Conjectures* Documentation! Check out the main [Formal Conjectures GitHub repository](https://github.com/google-deepmind/formal-conjectures) for more details. This page provides an overview of the problem categories and subject classifications used within the project. For a more detailed explanation of these categories and the AMS subject classifications, please refer to the [explanation of features in the project's README](https://github.com/google-deepmind/formal-conjectures?tab=readme-ov-file#some-features). --- ## Problem Category Statistics {categoryStats} (note the links above use GitHub search, and so require logging into GitHub) --- ## Subject Category Statistics {subjectStats} --- ## Repository growth " IO.println markdownBody let .some newBody := MD4Lean.renderHtml (parserFlags := MD4Lean.MD_FLAG_TABLES ) markdownBody | throwError "Parsing failed" let finalHtml ← replaceTag "main" inputHtmlContent (newBody ++ graphHtml) IO.FS.writeFile file finalHtml

def

docbuild

[ "import MD4Lean", "import Lean", "import Batteries.Data.String.Matcher", "import FormalConjectures.Util.Attributes", "import Mathlib.Data.String.Defs" ]

docbuild/scripts/overwrite_index.lean

main

null

IsSumDistinctSet (A : Finset ℕ) (N : ℕ) : Prop := A ⊆ Finset.Icc 1 N ∧ (fun (⟨S, _⟩ : A.powerset) => S.sum id).Injective /-- If $A\subseteq\{1, ..., N\}$ with $|A| = n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $$ N \gg 2 ^ n. $$ -/ @[category research open, AMS 5 11]

abbrev

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

IsSumDistinctSet

/-- A finite set of naturals $A$ is said to be a sum-distinct set for $N \in \mathbb{N}$ if $A\subseteq\{1, ..., N\}$ and the sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ -/

erdos_1 : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℕ) (_ : IsSumDistinctSet A N), N ≠ 0 → C * 2 ^ A.card < N := by sorry /-- The trivial lower bound is $N \gg 2^n / n$. -/ @[category undergraduate, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

erdos_1

/-- If $A\subseteq\{1, ..., N\}$ with $|A| = n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then $$ N \gg 2 ^ n. $$ -/

erdos_1.variants.weaker : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℕ) (_ : IsSumDistinctSet A N), N ≠ 0 → C * 2 ^ A.card / A.card < N := by sorry /-- Erdős and Moser [Er56] proved $$ N \geq (\tfrac{1}{4} - o(1)) \frac{2^n}{\sqrt{n}}. $$ [Er56] Erdős, P., _Problems and results in additive number theory_. Colloque sur la Th\'{E}orie des Nombres, Bruxelles, 1955 (1956), 127-137. -/ @[category research solved, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

erdos_1.variants.weaker

/-- The trivial lower bound is $N \gg 2^n / n$. -/

erdos_1.variants.lb : ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)), ∀ (N : ℕ) (A : Finset ℕ) (h : IsSumDistinctSet A N), (1 / 4 - o A.card) * 2 ^ A.card / (A.card : ℝ).sqrt ≤ N := by sorry /-- A number of improvements of the constant $\frac{1}{4}$ have been given, with the current record $\sqrt{2 / \pi}$ first provied in unpublished work of Elkies and Gleason. -/ @[category research solved, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

erdos_1.variants.lb

/-- Erdős and Moser [Er56] proved $$ N \geq (\tfrac{1}{4} - o(1)) \frac{2^n}{\sqrt{n}}. $$ [Er56] Erdős, P., _Problems and results in additive number theory_. Colloque sur la Th\'{E}orie des Nombres, Bruxelles, 1955 (1956), 127-137. -/

erdos_1.variants.lb_strong : ∃ (o : ℕ → ℝ) (_ : o =o[atTop] (1 : ℕ → ℝ)), ∀ (N : ℕ) (A : Finset ℕ) (h : IsSumDistinctSet A N), (√(2 / π) - o A.card) * 2 ^ A.card / (A.card : ℝ).sqrt ≤ N := by sorry /-- A finite set of real numbers is said to be sum-distinct if all the subset sums differ by at least $1$. -/

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

erdos_1.variants.lb_strong

/-- A number of improvements of the constant $\frac{1}{4}$ have been given, with the current record $\sqrt{2 / \pi}$ first provied in unpublished work of Elkies and Gleason. -/

IsSumDistinctRealSet (A : Finset ℝ) (N : ℕ) : Prop := A.toSet ⊆ Set.Ioc 0 N ∧ A.powerset.toSet.Pairwise fun S₁ S₂ => 1 ≤ dist (S₁.sum id) (S₂.sum id) /-- A generalisation of the problem to sets $A \subseteq (0, N]$ of real numbers, such that the subset sums all differ by at least $1$ is proposed in [Er73] and [ErGr80]. [Er73] Erdős, P., _Problems and results on combinatorial number theory_. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138. [ErGr80] Erdős, P. and Graham, R., _Old and new problems and results in combinatorial number theory_. Monographies de L'Enseignement Mathematique (1980). -/ @[category research open, AMS 5 11]

abbrev

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

IsSumDistinctRealSet

/-- A finite set of real numbers is said to be sum-distinct if all the subset sums differ by at least $1$. -/

erdos_1.variants.real : ∃ C > (0 : ℝ), ∀ (N : ℕ) (A : Finset ℝ) (_ : IsSumDistinctRealSet A N), N ≠ 0 → C * 2 ^ A.card < N := by sorry /-- The minimal value of $N$ such that there exists a sum-distinct set with three elements is $4$. https://oeis.org/A276661 -/ @[category undergraduate, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

erdos_1.variants.real

null

erdos_1.variants.least_N_3 : IsLeast { N | ∃ A, IsSumDistinctSet A N ∧ A.card = 3 } 4 := by refine ⟨⟨{1, 2, 4}, ?_⟩, ?_⟩ · simp refine ⟨by decide, ?_⟩ let P := Finset.powerset {1, 2, 4} have : Finset.univ.image (fun p : P ↦ ∑ x ∈ p, x) = {0, 1, 2, 4, 3, 5, 6, 7} := by refine Finset.ext_iff.mpr (fun n => ?_) simp [show P = {{}, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}} by decide] omega rw [Set.injective_iff_injOn_univ, ← Finset.coe_univ] have : (Finset.univ.image (fun p : P ↦ ∑ x ∈ p.1, x)).card = (Finset.univ (α := P)).card := by rw [this]; aesop exact Finset.injOn_of_card_image_eq this · simp [mem_lowerBounds] intro n S h h_inj hcard3 by_contra hn interval_cases n; aesop; aesop · have := Finset.card_le_card h aesop · absurd h_inj rw [(Finset.subset_iff_eq_of_card_le (Nat.le_of_eq (by rw [hcard3]; decide))).mp h] decide /-- The minimal value of $N$ such that there exists a sum-distinct set with five elements is $13$. https://oeis.org/A276661 -/ @[category research solved, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

erdos_1.variants.least_N_3

/-- The minimal value of $N$ such that there exists a sum-distinct set with three elements is $4$. https://oeis.org/A276661 -/

erdos_1.variants.least_N_5 : IsLeast { N | ∃ A, IsSumDistinctSet A N ∧ A.card = 5 } 13 := by sorry /-- The minimal value of $N$ such that there exists a sum-distinct set with nine elements is $161$. https://oeis.org/A276661 -/ @[category research solved, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

erdos_1.variants.least_N_5

/-- The minimal value of $N$ such that there exists a sum-distinct set with five elements is $13$. https://oeis.org/A276661 -/

erdos_1.variants.least_N_9 : IsLeast { N | ∃ A, IsSumDistinctSet A N ∧ A.card = 9 } 161 := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1.lean

erdos_1.variants.least_N_9

/-- The minimal value of $N$ such that there exists a sum-distinct set with nine elements is $161$. https://oeis.org/A276661 -/

sumPrimeAndTwoPows (k : ℕ) : Set ℕ := { p + (pows.map (2 ^ ·)).sum | (p : ℕ) (pows : Multiset ℕ) (_ : p.Prime) (_ : pows.card ≤ k)} /-- Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$? -/ @[category research open, AMS 5 11]

abbrev

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/10.lean

sumPrimeAndTwoPows

/-- The set of natural numbers that can be written as a sum of a prime and at most $k$ powers of $2$. -/

erdos_10 : answer(sorry) ↔ ∃ k, sumPrimeAndTwoPows k = Set.univ \ {0, 1} := by sorry /-- Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$ such that the set of integers which are the sum of a prime and at most $k(ϵ)$ many powers of $2$ has lower density at least $1 - ϵ$. Ref: Gallagher, P. X., _Primes and powers of 2_. -/ @[category research solved, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/10.lean

erdos_10

/-- Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of $2$? -/

erdos_10.variants.gallagher (ε : ℝ) (hε : 0 < ε) : ∃ k, 1 - ε ≤ (sumPrimeAndTwoPows k).lowerDensity := by sorry /-- Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. Ref: Granville, A. and Soundararajan, K., _A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$_ -/ @[category research open, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/10.lean

erdos_10.variants.gallagher

/-- Gallagher [Ga75] has shown that for any $ϵ > 0$ there exists $k(ϵ)$ such that the set of integers which are the sum of a prime and at most $k(ϵ)$ many powers of $2$ has lower density at least $1 - ϵ$. Ref: Gallagher, P. X., _Primes and powers of 2_. -/

erdos_10.variants.granville_soundararajan_odd : {n : ℕ | Odd n ∧ 1 < n} ⊆ sumPrimeAndTwoPows 3 ∧ {n : ℕ | Even n ∧ n ≠ 0} ⊆ sumPrimeAndTwoPows 4 := by sorry /-- Bogdan Grechuk has observed that `1117175146` is not the sum of a prime and at most $3$ powers of $2$. -/ @[category research solved, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/10.lean

erdos_10.variants.granville_soundararajan_odd

/-- Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of $2$ suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. Ref: Granville, A. and Soundararajan, K., _A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$_ -/

erdos_10.variants.grechuk_example : 1117175146 ∉ sumPrimeAndTwoPows 3 := by sorry /-- There are infinitely many even integers not the sum of a prime and $2$ powers of $2$ -/ @[category research solved, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/10.lean

erdos_10.variants.grechuk_example

/-- Bogdan Grechuk has observed that `1117175146` is not the sum of a prime and at most $3$ powers of $2$. -/

erdos_10.variants.two_pows : Set.Infinite <| {n : ℕ | Even n} \ sumPrimeAndTwoPows 2 := by sorry /-- Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$). -/ @[category research open, AMS 5 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/10.lean

erdos_10.variants.two_pows

/-- There are infinitely many even integers not the sum of a prime and $2$ powers of $2$ -/

erdos_10.variants.gretchuk : Set.Infinite <| {n : ℕ | Even n} \ sumPrimeAndTwoPows 3 := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/10.lean

erdos_10.variants.gretchuk

/-- Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$). -/

erdos_1003 : answer(sorry) ↔ Set.Infinite {n | φ n = φ (n + 1)} := by sorry /-- Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions. [Er85e] Erdős, P., _Some problems and results in number theory_. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. -/ @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1003.lean

erdos_1003

/-- Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function? -/

erdos_1003.variants.Icc : answer(sorry) ↔ ∀ k ≥ 1, {n | ∀ i ∈ Set.Icc 1 k, φ n = φ (n + i)}.Infinite := by sorry /-- Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$. [EPS87] Erd\H os, Paul and Pomerance, Carl and S\'ark\"ozy, Andr\'as, _On locally repeated values of certain arithmetic functions_. {II}. Proc. Amer. Math. Soc. (1987), 1--7. -/ @[category research solved, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1003.lean

erdos_1003.variants.Icc

/-- Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions. [Er85e] Erdős, P., _Some problems and results in number theory_. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87. -/

erdos_1003.variants.eps87 : ∀ᶠ x in atTop, {(n : ℕ) | (n ≤ x) ∧ φ n = φ (n + 1)}.ncard ≤ x / Real.exp ((x.log) ^ ((1 : ℝ) / 3)) := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1003.lean

erdos_1003.variants.eps87

/-- Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$. [EPS87] Erd\H os, Paul and Pomerance, Carl and S\'ark\"ozy, Andr\'as, _On locally repeated values of certain arithmetic functions_. {II}. Proc. Amer. Math. Soc. (1987), 1--7. -/

IsDistinctTotientRun (n K : ℕ) : Prop := (Set.Icc (n + 1) (n + K)).InjOn totient /-- For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem. -/ @[category research open, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1004.lean

IsDistinctTotientRun

/-- `IsDistinctTotientRun n K` means that the values `φ(n+1), φ(n+2), ..., φ(n+K)` are all distinct. -/

erdos_1004 : answer(sorry) ↔ ∀ c > (0 : ℝ), ∀ᶠ x in atTop, ∃ n ≤ x, IsDistinctTotientRun n ⌊(Real.log (x : ℝ)) ^ c⌋₊ := by sorry /-- Erdős, Pomerance, and Sárközy [EPS87] proved that if φ(n+k) are all distinct for 1 ≤ k ≤ K then K ≤ n / exp(c (log n)^{1/3}) for some constant c > 0. Here we state the existence of such a constant c. -/ @[category research solved, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1004.lean

erdos_1004

/-- For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c. This is an open problem. -/

erdos_1004.EPS87_theorem : answer(True) ↔ ∃ (c : ℝ) (hc : c > 0), ∀ (n K : ℕ), n > 0 → IsDistinctTotientRun n K → (K : ℝ) ≤ (n : ℝ) / Real.exp (c * (Real.log n) ^ (1/3 : ℝ)) := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1004.lean

erdos_1004.EPS87_theorem

/-- Erdős, Pomerance, and Sárközy [EPS87] proved that if φ(n+k) are all distinct for 1 ≤ k ≤ K then K ≤ n / exp(c (log n)^{1/3}) for some constant c > 0. Here we state the existence of such a constant c. -/

erdos_1038.inf (n : ℕ) : answer(sorry) = ⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧ (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree}, volume {x | |f.1.eval x| < 1} := by sorry /-- The infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `< 1.835`. -/ @[category research solved, AMS 28]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree" ]

FormalConjectures/ErdosProblems/1038.lean

erdos_1038.inf

/-- What is the infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]`? -/

erdos_1038.inf_upperBound (n : ℕ) : ⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧ (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree}, volume {x | |f.1.eval x| < 1} < 1.835 := by sorry /-- The infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `≥ 2 ^ (4 / 3) - 1`. -/ @[category research solved, AMS 28]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree" ]

FormalConjectures/ErdosProblems/1038.lean

erdos_1038.inf_upperBound

/-- The infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `< 1.835`. -/

erdos_1038.inf_lowerBound (n : ℕ) : 2 ^ (4 / 3 : ℝ) - 1 ≤ ⨅ f : {f : Polynomial ℝ // f.Monic ∧ f ≠ 1 ∧ (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree}, volume {x | |f.1.eval x| < 1} := by sorry /-- The supremum of `|{x ∈ ℝ : |f x| < 1}|` over all monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `2 * 2 ^ (1 / 2)`. This is proved in [Tao25]. -/ @[category research solved, AMS 28]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree" ]

FormalConjectures/ErdosProblems/1038.lean

erdos_1038.inf_lowerBound

/-- The infimum of `|{x ∈ ℝ : |f x| < 1}|` over all nonconstant monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `≥ 2 ^ (4 / 3) - 1`. -/

erdos_1038.sup (n : ℕ) : 2 * 2 ^ (1 / 2 : ℝ) = ⨆ f : {f : Polynomial ℝ // f.Monic ∧ (f.roots.filter fun x => x ∈ Set.Icc (-1 : ℝ) 1).card = f.natDegree}, volume {x | |f.1.eval x| < 1} := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree" ]

FormalConjectures/ErdosProblems/1038.lean

erdos_1038.sup

/-- The supremum of `|{x ∈ ℝ : |f x| < 1}|` over all monic polynomials `f` such that all of its roots are real and contained in `[-1,1]` is `2 * 2 ^ (1 / 2)`. This is proved in [Tao25]. -/

length (s : Set ℂ) : ℝ≥0∞ := μH[1] s /-- **Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958): If $f$ is a monic degree $n$ polynomial with all roots in the unit disk, then some connected component of $\{z \mid |f(z)| < 1\}$ contains at least two roots with multiplicity. See p. 139, above Problem 5: [EHP58] Erdős, P. and Herzog, F. and Piranian, G., _Metric properties of polynomials_. J. Analyse Math. (1958), 125-148. -/ @[category research solved, AMS 32]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1041.lean

length

/-- The length of a subset $s$ of $\mathbb{C}$ is defined to be its 1-dimensional Hausdorff measure $\mathcal{H}^1(s)$. -/

exists_connected_component_contains_two_roots : ∃ C, C ⊆ {z | ‖f.eval z‖ < 1} ∧ IsConnected C ∧ 2 ≤ (f.roots.filter (· ∈ C)).card := by sorry /-- Let $$ f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x] $$ with $|z_i| < 1$ for all $i$. Conjecture: Must there always exist a path of length less than 2 in $$ \{ z \in \mathbb{C} \mid |f(z)| < 1 \} $$ which connects two of the roots of $f$? -/ @[category research open, AMS 32]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1041.lean

exists_connected_component_contains_two_roots

/-- **Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958): If $f$ is a monic degree $n$ polynomial with all roots in the unit disk, then some connected component of $\{z \mid |f(z)| < 1\}$ contains at least two roots with multiplicity. See p. 139, above Problem 5: [EHP58] Erdős, P. and Herzog, F. and Piranian, G., _Metric properties of polynomials_. J. Analyse Math. (1958), 125-148. -/

erdos_1041 : ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂), Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ length (Set.range γ) < 2 := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1041.lean

erdos_1041

/-- Let $$ f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x] $$ with $|z_i| < 1$ for all $i$. Conjecture: Must there always exist a path of length less than 2 in $$ \{ z \in \mathbb{C} \mid |f(z)| < 1 \} $$ which connects two of the roots of $f$? -/

levelSet (f : Polynomial ℂ) : Set ℂ := {z : ℂ | ‖f.eval z‖ ≤ 1} /-- **Erdős Problem 1043**: Let $f\in \mathbb{C}[x]$ be a monic polynomial. Must there exist a straight line $\ell$ such that the projection of \[\{ z: \lvert f(z)\rvert\leq 1\}\] onto $\ell$ has measure at most $2$? Pommerenke [Po61] proved that the answer is no. [Po61] Pommerenke, Ch., _On metric properties of complex polynomials._ Michigan Math. J. (1961), 97-115. -/ @[category research solved, AMS 28 30]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1043.lean

levelSet

/-- The set $\{ z \in \mathbb{C} : \lvert f(z)\rvert\leq 1\}$ -/

erdos_1043 : answer(False) ↔ ∀ (f : ℂ[X]), f.Monic → f.degree ≥ 1 → ∃ (u : ℂ), ‖u‖ = 1 ∧ volume ((ℝ ∙ u).orthogonalProjection '' levelSet f) ≤ 2 := by sorry /-- On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most 3.3. -/ @[category research solved, AMS 28 30]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1043.lean

erdos_1043

/-- **Erdős Problem 1043**: Let $f\in \mathbb{C}[x]$ be a monic polynomial. Must there exist a straight line $\ell$ such that the projection of \[\{ z: \lvert f(z)\rvert\leq 1\}\] onto $\ell$ has measure at most $2$? Pommerenke [Po61] proved that the answer is no. [Po61] Pommerenke, Ch., _On metric properties of complex polynomials._ Michigan Math. J. (1961), 97-115. -/

erdos_1043.variants.weak : ∀ (f : ℂ[X]), f.Monic → f.degree ≥ 1 → ∃ (u : ℂ), ‖u‖ = 1 ∧ volume ((ℝ ∙ u).orthogonalProjection '' levelSet f) ≤ 3.3 := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1043.lean

erdos_1043.variants.weak

/-- On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most 3.3. -/

GrowthCondition (a : ℕ → ℤ) : Prop := Filter.liminf (fun n => ((a n : ℝ) ^ (1 / 2 ^ n : ℝ))) Filter.atTop > 1 /-- The series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$. -/

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1051.lean

GrowthCondition

/-- A sequence of integers `a` satisfies the growth condition if $\liminf a_n^{\frac{1}{2^n}} > 1$. -/

ErdosSeries (a : ℕ → ℤ) : ℝ := ∑' n : ℕ, 1 / ((a n : ℝ) * (a (n + 1) : ℝ)) /-- Is it true that if $a_0 < a_1 < a_2 < \cdots$ is a strictly increasing sequence of integers with $\liminf a_n^{1/2^n} > 1$, then the series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$ is irrational? -/ @[category research open, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1051.lean

ErdosSeries

/-- The series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$. -/

erdos_1051 : answer(sorry) ↔ ∀ (a : ℕ → ℤ), StrictMono a → GrowthCondition a → Irrational (ErdosSeries a) := by sorry /-- Erdős [Er88c] notes that if the sequence grows rapidly to infinity (specifically, if $a_{n+1} \geq C \cdot a_n^2$ for some constant $C > 0$), then the series is irrational. [Er88c] Erdős, P., _On the irrationality of certain series: problems and results_. New advances in transcendence theory (Durham, 1986) (1988), 102-109. -/ @[category research solved, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1051.lean

erdos_1051

/-- Is it true that if $a_0 < a_1 < a_2 < \cdots$ is a strictly increasing sequence of integers with $\liminf a_n^{1/2^n} > 1$, then the series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$ is irrational? -/

erdos_1051.rapid_growth (a : ℕ → ℤ) (h_mono : StrictMono a) (h_rapid : ∃ C > 0, ∀ n, (a (n + 1) : ℝ) ≥ C * (a n : ℝ) ^ 2) : Irrational (ErdosSeries a) := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1051.lean

erdos_1051.rapid_growth

/-- Erdős [Er88c] notes that if the sequence grows rapidly to infinity (specifically, if $a_{n+1} \geq C \cdot a_n^2$ for some constant $C > 0$), then the series is irrational. [Er88c] Erdős, P., _On the irrationality of certain series: problems and results_. New advances in transcendence theory (Durham, 1986) (1988), 102-109. -/

properUnitaryDivisors (n : ℕ) : Finset ℕ := {d ∈ Finset.Ico 1 n | d ∣ n ∧ d.Coprime (n / d)} /-- A number $n > 0$ is a unitary perfect number if it is the sum of its proper unitary divisors. -/

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

properUnitaryDivisors

/-- A proper unitary divisor of $n$ is a divisor $d$ of $n$ such that $d$ is coprime to $n/d$, and $d < n$. -/

IsUnitaryPerfect (n : ℕ) : Prop := ∑ i ∈ properUnitaryDivisors n, i = n ∧ 0 < n /-- Are there only finitely many unitary perfect numbers? -/ @[category research open, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

IsUnitaryPerfect

/-- A number $n > 0$ is a unitary perfect number if it is the sum of its proper unitary divisors. -/

erdos_1052 : answer(sorry) ↔ {n | IsUnitaryPerfect n}.Finite := by sorry /-- All unitary perfect numbers are even. -/ @[category research solved, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

erdos_1052

/-- Are there only finitely many unitary perfect numbers? -/

even_of_isUnitaryPerfect (n : ℕ) (hn : IsUnitaryPerfect n) : Even n := by sorry @[category test, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

even_of_isUnitaryPerfect

/-- All unitary perfect numbers are even. -/

isUnitaryPerfect_6 : IsUnitaryPerfect 6 := by norm_num [IsUnitaryPerfect, properUnitaryDivisors] decide +kernel @[category test, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

isUnitaryPerfect_6

/-- All unitary perfect numbers are even. -/

isUnitaryPerfect_60 : IsUnitaryPerfect 60 := by norm_num [IsUnitaryPerfect, properUnitaryDivisors] decide +kernel @[category test, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

isUnitaryPerfect_60

/-- All unitary perfect numbers are even. -/

isUnitaryPerfect_90 : IsUnitaryPerfect 90 := by norm_num [IsUnitaryPerfect, properUnitaryDivisors] decide +kernel @[category test, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

isUnitaryPerfect_90

/-- All unitary perfect numbers are even. -/

isUnitaryPerfect_87360 : IsUnitaryPerfect 87360 := by norm_num [IsUnitaryPerfect, properUnitaryDivisors] decide +kernel @[category test, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

isUnitaryPerfect_87360

null

isUnitaryPerfect_146361946186458562560000 : IsUnitaryPerfect 146361946186458562560000 := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1052.lean

isUnitaryPerfect_146361946186458562560000

null

f (n : ℕ) : ℕ := if h : ∃ᵉ (m) (k ≥ 1), n = ∑ i < k, Nat.nth (· ∈ m.divisors) i then Nat.find h else 0 /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?-/ @[category research open, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1054.lean

f

/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$.-/

erdos_1054.parts.i : answer(sorry) ↔ (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := by sorry /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$? -/ @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1054.lean

erdos_1054.parts.i

/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$?-/

erdos_1054.parts.ii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧ (fun (n : A) ↦ (f ↑n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ)) := by sorry /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$? -/ @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1054.lean

erdos_1054.parts.ii

/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $f(n)=o(n)$ for almost all $n$? -/

erdos_1054.parts.iii : answer(sorry) ↔ ∃ (A : Set ℕ), A.HasDensity 1 ∧ atTop.limsup (fun n ↦ (f n : EReal) / n) = ⊤ := by sorry /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$. -/ @[category high_school, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1054.lean

erdos_1054.parts.iii

/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Is it true that $\limsup f(n)/n=\infty$? -/

f_undefined_at_2 : f 2 = 0 := by sorry /-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$. -/ @[category high_school, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1054.lean

f_undefined_at_2

/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=2$, i.e. we get the junk value $0$. -/

f_undefined_at_3 : f 5 = 0 := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1054.lean

f_undefined_at_3

/-- Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$. Show that $f$ is undefined at $n=5$, i.e. we get the junk value $0$. -/

IsOfClass : ℕ+ → ℕ → Prop := fun r ↦ PNat.caseStrongInductionOn (p := fun (_ : ℕ+) ↦ ℕ → Prop) r (fun p ↦ (p + 1).primeFactors ⊆ {2, 3}) (fun n H p ↦ (∀ r ∈ (p + 1).primeFactors, ∃ (m : ℕ+) (hm : m ≤ n), H m hm r) ∧ (∃ r ∈ (p + 1).primeFactors, ∀ (m : ℕ+) (hm : m ≤ n), H m hm r → m = n)) /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Show that for each $r$ there exists a prime $p$ of class $r$. -/ @[category undergraduate, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1055.lean

IsOfClass

/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor.-/

exists_p (r : ℕ+) : ∃ p, p.Prime ∧ IsOfClass r p := by sorry open Classical /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Let $p_r$ is the least prime in class $r$.-/

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1055.lean

exists_p

/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Show that for each $r$ there exists a prime $p$ of class $r$. -/

p (r : ℕ+) : ℕ := Nat.find (exists_p r) /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Are there infinitely many primes in each class?-/ @[category research open, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1055.lean

p

/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Let $p_r$ is the least prime in class $r$.-/

erdos_1055 (r) : {p | p.Prime ∧ IsOfClass r p}.Infinite := by sorry /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Erdos conjectured that this tends to infinity. -/ @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1055.lean

erdos_1055

/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Are there infinitely many primes in each class?-/

erdos_1055.variants.erdos_limit : Filter.atTop.Tendsto (fun r ↦ (p r : ℝ) ^ (1 / r : ℝ)) Filter.atTop := by sorry /-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Selfridge conjectured that this is bounded. -/ @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1055.lean

erdos_1055.variants.erdos_limit

/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Erdos conjectured that this tends to infinity. -/

erdos_1055.variants.selfridge_limit : ∃ M, ∀ r, (p r : ℝ) ^ (1 / r : ℝ) ≤ M := by sorry --TODO(Paul-Lez): formalize the rest of the problems on the page.

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1055.lean

erdos_1055.variants.selfridge_limit

/-- A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Selfridge conjectured that this is bounded. -/

AllModProdEqualsOne (p : ℕ) {k : ℕ} (boundaries : Fin (k + 1) → ℕ) : Prop := ∀ i : Fin k, (∏ n ∈ Finset.Ico (boundaries i.castSucc) (boundaries (i.castSucc + 1)), n) ≡ 1 [MOD p] /-- Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$ such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$? -/ @[category research open, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1056.lean

AllModProdEqualsOne

/-- The proposition that the modular product of a collection of consecutive interval equals $1$ modulo $p$, where intervals are defined by a function specifying the consecutive boundaries. -/

erdos_1056 : answer(sorry) ↔ ∀ k ≥ 2, ∃ (p : ℕ) (_ : p.Prime) (boundaries : Fin (k + 1) → ℕ) (_ : StrictMono boundaries), AllModProdEqualsOne p boundaries := by sorry /-- This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979 Erdős observed that $3 * 4 \equiv 5 * 6 * 7 \equiv 1 \mod 11$. -/ @[category undergraduate, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1056.lean

erdos_1056

/-- Let $k ≥ 2$. Does there exist a prime $p$ and consecutive intervals $I_0,\dots,I_k$ such that $\prod\limits_{n{\in}I_i}n \equiv 1 \mod n$ for all $1 \le i \le k$? -/

erdos_1056_k2 : AllModProdEqualsOne 11 ![3, 5, 8] := by unfold AllModProdEqualsOne decide /-- Makowski [Ma83] found, for $k=3$: $2 * 3 * 4 * 5 \equiv 6 * 7 * 8 * 9 * 10 * 11 \equiv 12 * 13 * 14 * 15 \equiv 1 \mod 17$. -/ @[category undergraduate, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1056.lean

erdos_1056_k2

/-- This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979 Erdős observed that $3 * 4 \equiv 5 * 6 * 7 \equiv 1 \mod 11$. -/

erdos_1056_k3 : AllModProdEqualsOne 17 ![2, 6, 12, 16] := by unfold AllModProdEqualsOne decide /-- Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$). -/ @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1056.lean

erdos_1056_k3

/-- Makowski [Ma83] found, for $k=3$: $2 * 3 * 4 * 5 \equiv 6 * 7 * 8 * 9 * 10 * 11 \equiv 12 * 13 * 14 * 15 \equiv 1 \mod 17$. -/

noll_simmons : answer(sorry) ↔ ∀ᶠ k in Filter.atTop, ∃ (p : ℕ) (_ : p.Prime) (Q : Fin k → ℕ) (_ : StrictMono Q) (_ : ∀ i, Q i < p), ∀ i j : Fin k, (Q i)! ≡ (Q j)! [MOD p] := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1056.lean

noll_simmons

/-- Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$). -/

IsFactorial (d : ℕ) : Prop := d ∈ Set.range Nat.factorial

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

IsFactorial

null

factorialsLessThanN (n : ℕ) : Set ℕ := { d | d < n ∧ IsFactorial d }

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

factorialsLessThanN

null

AllFactorialSubtractionsComposite (n : ℕ) : Prop := ∀d ∈ factorialsLessThanN n, (n - d).Composite /-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/ @[category research open, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

AllFactorialSubtractionsComposite

null

erdos_1059 : answer(sorry) ↔ Set.Infinite {p | p.Prime ∧ AllFactorialSubtractionsComposite p} := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

erdos_1059

/-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/

DecidableIsFactorial (d : ℕ) : Prop := ((Finset.Icc 0 d).filter (λ k => Nat.factorial k = d)).Nonempty

abbrev

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

DecidableIsFactorial

/-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/

decidableFactorialsLessThanN (n : ℕ) : Finset ℕ := (Finset.range n).filter DecidableIsFactorial

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

decidableFactorialsLessThanN

/-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/

DecidableAllFactorialSubtractionsComposite (n : ℕ) : Prop := ∀ d ∈ decidableFactorialsLessThanN n, (n - d).Composite @[category test, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

DecidableAllFactorialSubtractionsComposite

/-- Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$? -/

isFactorial_equivalent (d : ℕ) : IsFactorial d ↔ DecidableIsFactorial d := by unfold IsFactorial DecidableIsFactorial simp constructor · rintro ⟨k, hk⟩ use k rw [Finset.mem_filter] constructor · have hk : k <= d := by rw [← hk] apply Nat.self_le_factorial rw [Finset.mem_Icc] exact ⟨Nat.zero_le k, hk⟩ · exact hk · rintro ⟨k, hk⟩ use k rw [Finset.mem_filter] at hk exact hk.2 @[category test, AMS 11]

lemma

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

isFactorial_equivalent

null

factorialsLessThanN_equivalent (n : ℕ) : factorialsLessThanN n = ↑(decidableFactorialsLessThanN n) := by ext d unfold factorialsLessThanN decidableFactorialsLessThanN simp exact λ _ => isFactorial_equivalent d @[category test, AMS 11]

lemma

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

factorialsLessThanN_equivalent

null

allFactorialSubtractionsComposite_equivalent (d : ℕ) : DecidableAllFactorialSubtractionsComposite d ↔ AllFactorialSubtractionsComposite d := by unfold AllFactorialSubtractionsComposite DecidableAllFactorialSubtractionsComposite rw [factorialsLessThanN_equivalent d] simp @[category test, AMS 11]

lemma

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

allFactorialSubtractionsComposite_equivalent

null

allFactorialSubtractionsComposite_101 : AllFactorialSubtractionsComposite 101 := by have h : DecidableAllFactorialSubtractionsComposite 101 := by norm_num [DecidableAllFactorialSubtractionsComposite, decidableFactorialsLessThanN] decide +kernel exact (allFactorialSubtractionsComposite_equivalent 101).mp h @[category test, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

allFactorialSubtractionsComposite_101

null

allFactorialSubtractionsComposite_211 : AllFactorialSubtractionsComposite 211 := by have h : DecidableAllFactorialSubtractionsComposite 211 := by norm_num [DecidableAllFactorialSubtractionsComposite, decidableFactorialsLessThanN] decide +kernel exact (allFactorialSubtractionsComposite_equivalent 211).mp h @[category test, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

allFactorialSubtractionsComposite_211

null

notAllFactorialSubtractionsComposite_89 : ¬(AllFactorialSubtractionsComposite 89) := by have h : ¬(DecidableAllFactorialSubtractionsComposite 89) := by unfold DecidableAllFactorialSubtractionsComposite decidableFactorialsLessThanN intro h specialize h 6 have : Nat.Prime (89 - 6) := by norm_num contradiction simp [allFactorialSubtractionsComposite_equivalent] at h exact h @[category test, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

notAllFactorialSubtractionsComposite_89

null

testFactorialsLessThanN : factorialsLessThanN 100 = {1, 2, 6, 24} := by have h : decidableFactorialsLessThanN 100 = {1, 2, 6, 24} := by norm_num [decidableFactorialsLessThanN] decide +kernel rw [factorialsLessThanN_equivalent] simp [h]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1059.lean

testFactorialsLessThanN

null

erdos_1060.bound_one : ∃ h : ℕ → ℝ, h =o[atTop] (fun n ↦ 1 / log (log n)) ∧ ∀ᶠ n in atTop, #{k ≤ n | k * σ 1 k = n} ≤ (n : ℝ) ^ h n := by sorry @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1060.lean

erdos_1060.bound_one

/-- The conjecture is about the function $f(n)$ which counts the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$. The first bound is that $f(n)$ grows slower than any power of $n^(\frac{1}{\log\log n})$. The second bound is that $f(n)$ is at most a power of $\log n$. -/

erdos_1060.bound_two : ∃ (C : ℝ), (fun n ↦ (#{k ≤ n | k * σ 1 k = n} : ℝ)) =O[atTop] (fun n ↦ log n ^ C) := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1060.lean

erdos_1060.bound_two

null

ForkFree (A : Set ℕ) : Prop := ∀ a ∈ A, ({b | b ∈ A \ {a} ∧ a ∣ b} : Set ℕ).Subsingleton open scoped Classical in /-- The extremal function from Erdős problem 1062: the largest size of a fork-free subset of `{1,...,n}`. -/

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]

FormalConjectures/ErdosProblems/1062.lean

ForkFree

/-- A set `A` of positive integers is fork-free if no element divides two distinct other elements of `A`. -/

f (n : ℕ) : ℕ := Nat.findGreatest (fun k => ∃ A ⊆ Set.Icc 1 n, ForkFree A ∧ A.ncard = k) n /-- The interval `[⌊n/3⌋, n]` is fork-free, and therefore `f n` is at least `⌈2n / 3⌉`. -/ @[category research solved, AMS 11]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]

FormalConjectures/ErdosProblems/1062.lean

f

/-- The extremal function from Erdős problem 1062: the largest size of a fork-free subset of `{1,...,n}`. -/

erdos_1062.lower_bound (n : ℕ) : ⌈(2 * n / 3 : ℝ)⌉₊ ≤ f n := by classical set b : ℕ := n / 3 with hb let A : Finset ℕ := .Icc (b + 1) n calc ⌈(2 * n / 3 : ℝ)⌉₊ ≤ n - b := by grw [Nat.ceil_le, Nat.cast_sub (by omega), le_sub_iff_add_le, hb, Nat.cast_div_le] -- FIXME: `ring` should have some basic inequality support. apply le_of_eq ring _ ≤ f n := Nat.le_findGreatest (by omega) ⟨A, by simp only [Finset.coe_Icc, A]; gcongr; omega, ?_, by simp [A, -Finset.coe_Icc]⟩ simp only [ForkFree, Finset.coe_Icc, Set.mem_Icc, Set.mem_diff, Set.mem_singleton_iff, and_assoc, and_imp, A] rintro a ha - refine Set.subsingleton_of_forall_eq (a * 2) ?_ simp only [Set.mem_setOf_eq, and_imp] rintro _ _ hk _ ⟨k, rfl⟩ match k with | 0 | 1 | 2 => simp_all | k + 3 => grw [← le_add_self] at hk; omega /-- Lebensold proved that for large `n`, the function `f n` lies between `0.6725 n` and `0.6736 n`. -/ @[category research solved, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]

FormalConjectures/ErdosProblems/1062.lean

erdos_1062.lower_bound

/-- The interval `[⌊n/3⌋, n]` is fork-free, and therefore `f n` is at least `⌈2n / 3⌉`. -/

erdos_1062.lebensold_bounds : ∀ᶠ n in atTop, (0.6725 : ℝ) * n ≤ f n ∧ f n ≤ (0.6736 : ℝ) * n := by sorry /-- Erdős asked whether the limiting density `f n / n` exists and, if so, whether it is irrational. -/ @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]

FormalConjectures/ErdosProblems/1062.lean

erdos_1062.lebensold_bounds

/-- Lebensold proved that for large `n`, the function `f n` lies between `0.6725 n` and `0.6736 n`. -/

erdos_1062.limit_density : (∃ l, Tendsto (fun n => (f n : ℝ) / n) atTop (𝓝 l) ∧ Irrational l) ↔ answer(sorry) := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports", "import Mathlib.Topology.Basic" ]

FormalConjectures/ErdosProblems/1062.lean

erdos_1062.limit_density

/-- Erdős asked whether the limiting density `f n / n` exists and, if so, whether it is irrational. -/

erdos_1064 : {n | φ n > φ (n - φ n)}.HasDensity 1 := by sorry /-- Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$ such that $ϕ(n)< ϕ(n - ϕ(n))$ Reference: [GLW01] Grytczuk, A. and Luca, F. and W\'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the {E}uler totient function. -/ @[category research solved, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1064.lean

erdos_1064

/-- Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$ have asymptotic density 1. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. -/

erdos_1064.variants.k2 : {n | φ n < φ (n - φ n)}.Infinite := by sorry open Asymptotics Filter /-- For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. (2002), 111--130. -/ @[category research solved, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1064.lean

erdos_1064.variants.k2

/-- Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$ such that $ϕ(n)< ϕ(n - ϕ(n))$ Reference: [GLW01] Grytczuk, A. and Luca, F. and W\'ojtowicz, M., A conjecture of {E}rdős concerning inequalities for the {E}uler totient function. -/

erdos_1064.variants.general_function (f : ℕ → ℕ) (hf : (fun n ↦ (f n : ℝ)) =o[atTop] (fun n ↦ (n : ℝ))) : {n : ℕ | φ (n - φ n) + f n < φ n}.HasDensity 1 := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1064.lean

erdos_1064.variants.general_function

/-- For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$. Reference: [LuPo02] Luca, Florian and Pomerance, Carl, On some problems of {M}\polhk akowski-{S}chinzel and {E}rd\H os concerning the arithmetical functions {$\phi$} and {$\sigma$}. Colloq. Math. (2002), 111--130. -/

erdos_1065a : answer(sorry) ↔ Set.Infinite {p | ∃ q k, p.Prime ∧ q.Prime ∧ p = 2^k * q + 1} := by sorry /-- Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$ for some prime $q$ and $k ≥ 0$, $l ≥ 0$? -/ @[category research open, AMS 11]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1065.lean

erdos_1065a

/-- Are there infinitely many primes $p$ such that $p = 2^k * q + 1$ for some prime $q$ and $k ≥ 0$? This is mentioned as B46 in [Unsolved Problems in Number Theory](https://doi.org/10.1007/978-0-387-26677-0) by *Richard K. Guy* -/

erdos_1065b : answer(sorry) ↔ Set.Infinite {p | ∃ q k l, p.Prime ∧ q.Prime ∧ p = 2^k * 3^l * q + 1} := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1065.lean

erdos_1065b

/-- Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$ for some prime $q$ and $k ≥ 0$, $l ≥ 0$? -/

InternallyDisjoint {V : Type*} {G : SimpleGraph V} {u v x y : V} (p : G.Walk u v) (q : G.Walk x y) : Prop := Disjoint p.support.tail.dropLast q.support.tail.dropLast /-- We say a graph is infinitely connected if any two vertices are connected by infinitely many pairwise disjoint paths. -/

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1068.lean

InternallyDisjoint

/-- Two walks are internally disjoint if they share no vertices other than their endpoints. -/

InfinitelyConnected {V : Type*} (G : SimpleGraph V) : Prop := Pairwise fun u v ↦ ∃ P : Set (G.Walk u v), P.Infinite ∧ (∀ p ∈ P, p.IsPath) ∧ P.Pairwise InternallyDisjoint /-- Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected? -/ @[category research open, AMS 5]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1068.lean

InfinitelyConnected

/-- We say a graph is infinitely connected if any two vertices are connected by infinitely many pairwise disjoint paths. -/

erdos_1068 : answer(sorry) ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticNumber = aleph 1 → ∃ s : Set V, s.Countable ∧ InfinitelyConnected (G.induce s) := by sorry

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/1068.lean

erdos_1068

/-- Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected? -/

cardSet (n : ℕ) := { N | ∀ (pts : Finset ℝ²), pts.card = N → NonTrilinear pts.toSet → HasConvexNGon n pts } /-- The function $f(n)$ specified in `erdos_107`. -/

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/107.lean

cardSet

/-- The set of $N$ such that any $N$ points in the plane, no three on a line, contain a convex $n$-gon. -/

f (n : ℕ) : ℕ := sInf (cardSet n) /-- Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n) = 2^{n-2} + 1$. -/ @[category research open, AMS 52]

def

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/107.lean

f

/-- The function $f(n)$ specified in `erdos_107`. -/

erdos_107 : answer(sorry) ↔ ∀ n ≥ 3, f n = 2^(n - 2) + 1 := by sorry /-- For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line, contain a convex $n$-gon. -/ @[category research solved, AMS 52]

theorem

FormalConjectures

[ "import FormalConjectures.Util.ProblemImports" ]

FormalConjectures/ErdosProblems/107.lean

erdos_107

/-- Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n) = 2^{n-2} + 1$. -/

Datasets:

phanerozoic
/

Lean4-FormalConjectures

Lean4-FormalConjectures

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Column	Type	Description
fact	string	Declaration body
type	string	theorem, def, lemma, etc.
library	string	Source module
imports	list	Required imports
filename	string	Source file path
symbolic_name	string	Identifier
docstring	string	Documentation (if present)