Datasets:

saidutta69
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unsolved-math-clean

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MPP-001	P versus NP Problem	Does $P = NP$? More formally: if the solution to a problem can be quickly verified (in polynomial time), can the solution also be quickly found (in polynomial time)?	The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be quickly verified can also be quickly solved. The Clay Mathematics Institute has offered a $1,000,000 prize for a correct solution. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked ...	open	L5: Millennium Prize	5	Computer Science	Millennium Prize Problems	Stephen Cook	1,971	1,523	89	train
MPP-002	The Riemann Hypothesis	Do all non-trivial zeros of the Riemann zeta function $\zeta(s)$ have real part equal to $\frac{1}{2}$?	The Riemann hypothesis, proposed by Bernhard Riemann in 1859, concerns the distribution of prime numbers. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line $\Re(s) = \frac{1}{2}$. This is one of the most important open problems in mathematics, with profound implications for numb...	open	L5: Millennium Prize	5	Number Theory	Millennium Prize Problems	Bernhard Riemann	1,859	2,341	156	train
MPP-003	Yang–Mills Existence and Mass Gap	Prove that Yang–Mills theory exists and has a mass gap on $\mathbb{R}^4$, meaning the quantum particles have positive masses.	This problem concerns quantum field theory and seeks to establish a rigorous mathematical foundation for Yang–Mills theories, which describe fundamental forces in particle physics. A solution would require proving the existence of these theories in four-dimensional spacetime and showing they predict a mass gap. <!-- L...	open	L5: Millennium Prize	5	Mathematical Physics	Millennium Prize Problems	Yang Chen-Ning and Robert Mills	1,954	1,234	78	train
MPP-004	Navier–Stokes Existence and Smoothness	Prove or give a counterexample: Do solutions to the Navier–Stokes equations in three dimensions always exist and remain smooth for all time?	The Navier–Stokes equations describe the motion of fluids. While solutions exist for short times and in two dimensions, the question of whether smooth solutions exist globally in three dimensions remains open. This has profound implications for understanding turbulence and fluid dynamics. <!-- LITERATURE-TRIAGE:BEGIN ...	open	L5: Millennium Prize	5	Partial Differential Equations	Millennium Prize Problems	Claude-Louis Navier and George Gabriel Stokes	1,822	1,456	89	train
MPP-005	Birch and Swinnerton-Dyer Conjecture	The conjecture relates the rank of the abelian group of rational points of an elliptic curve to the order of zero of the associated L-function at $s=1$.	This conjecture connects the arithmetic of elliptic curves (solutions to equations of the form $y^2 = x^3 + ax + b$) to the behavior of certain complex functions. It has deep connections to number theory and algebraic geometry. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: ope...	open	L5: Millennium Prize	5	Number Theory	Millennium Prize Problems	Bryan Birch and Peter Swinnerton-Dyer	1,960	1,123	67	train
NT-001	Odd Perfect Numbers	Does there exist an odd perfect number? A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself). For example, $6 = 1 + 2 + 3$ is perfect.	While many even perfect numbers are known (the first few are 6, 28, 496, 8128), no odd perfect number has ever been found, despite extensive computer searches. It has been proven that if one exists, it must be greater than $10^{1500}$ and have at least 101 prime factors. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature ...	open	L3: Advanced	3	Number Theory	null	null	null	543	34	train
NT-002	Collatz Conjecture	Starting with any positive integer $n$, repeatedly apply the function: if $n$ is even, divide by 2; if $n$ is odd, multiply by 3 and add 1. Does this process always eventually reach 1?	Also known as the 3n+1 problem, this deceptively simple conjecture has been verified for all starting values up to $2^{68}$ but remains unproven. Paul Erdős said about it: "Mathematics may not be ready for such problems." <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially...	open	L4: Expert	4	Number Theory	null	null	null	892	67	train
NT-004	Goldbach's Conjecture	Every even integer greater than 2 can be expressed as the sum of two primes.	Proposed by Christian Goldbach in 1742, this conjecture has been verified computationally for all even integers up to very large numbers. The weak Goldbach conjecture (every odd number greater than 5 is the sum of three primes) was proved by Harald Helfgott in 2013, but the strong version remains open. <!-- LITERATURE...	open	L4: Expert	4	Number Theory	null	Christian Goldbach	1,742	1,567	112	train
NT-005	ABC Conjecture	For any $\epsilon > 0$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers with $a + b = c$ such that $c > \text{rad}(abc)^{1+\epsilon}$, where $\text{rad}(n)$ is the product of distinct prime factors of $n$.	The ABC conjecture, formulated by Joseph Oesterlé and David Masser in 1985, has profound implications for number theory. Shinichi Mochizuki claimed a proof in 2012 using his "inter-universal Teichmüller theory," but the proof remains controversial and not widely accepted. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature...	open	L5: Millennium Prize	5	Number Theory	null	Joseph Oesterlé and David Masser	1,985	876	45	train
COMB-001	The Hadwiger-Nelson Problem	What is the minimum number of colors needed to color the points of the plane such that no two points at distance 1 have the same color?	It is known that this chromatic number is between 5 and 7. In 2018, Aubrey de Grey proved it is at least 5, but whether it is 5, 6, or 7 remains unknown. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved Classification: PARTIAL-PROGRESS **Current literature...	open	L3: Advanced	3	Combinatorics	null	null	null	421	28	train
GT-001	Hadwiger Conjecture	Every graph with chromatic number $k$ has a $K_k$ minor (where $K_k$ is the complete graph on $k$ vertices).	The Hadwiger conjecture, proposed in 1943, generalizes the four color theorem. It has been proved for $k \leq 6$ but remains open for $k \geq 7$. The case $k=5$ is equivalent to the four color theorem. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved **Classifi...	open	L4: Expert	4	Graph Theory	null	Hugo Hadwiger	1,943	654	38	train
GT-002	Reconstruction Conjecture	Every finite simple graph on at least 3 vertices is uniquely determined by its vertex-deleted subgraphs.	The reconstruction conjecture asks whether a graph can be uniquely reconstructed from the multiset of all its vertex-deleted subgraphs. Proposed by Stanisław Ulam in 1942, it has been verified for many classes of graphs but remains open in general. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08...	open	L3: Advanced	3	Graph Theory	null	Stanisław Ulam	1,942	432	24	train
GEO-002	Sphere Packing in Higher Dimensions	What is the densest packing of congruent spheres in $n$ dimensions for $n \geq 4$?	The sphere packing problem asks for the densest arrangement of non-overlapping spheres. Maryna Viazovska solved it for dimension 8 in 2016, and she with collaborators solved it for dimension 24 in 2017. The problem remains open for most other dimensions. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2...	open	L4: Expert	4	Geometry	null	null	null	456	27	train
ALG-001	Inverse Galois Problem	Is every finite group the Galois group of some Galois extension of the rational numbers $\mathbb{Q}$?	The inverse Galois problem asks whether every finite group can be realized as the Galois group of a polynomial equation with rational coefficients. It has been solved for many classes of groups, including all symmetric and alternating groups, but remains open in general. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature ...	open	L4: Expert	4	Algebra	null	null	null	543	29	train
ALG-002	Kaplansky's Conjectures	A set of conjectures about group rings: (1) Zero divisor conjecture: If $G$ is a torsion-free group and $K$ is a field, then $K[G]$ has no zero divisors. (2) Idempotent conjecture: The only idempotents in $K[G]$ are 0 and 1. (3) Unit conjecture: The only units in $\mathbb{Z}[G]$ are of the form $\pm g$ for $g \in G$.	These conjectures, proposed by Irving Kaplansky in the 1940s, concern the algebraic structure of group rings. They have been verified for many classes of groups but remain open in general. The zero divisor conjecture is related to the Atiyah conjecture. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 20...	open	L4: Expert	4	Algebra	null	Irving Kaplansky	1,940	321	18	train
SET-001	Continuum Hypothesis	There is no set whose cardinality is strictly between that of the integers and the real numbers.	The continuum hypothesis was the first of Hilbert's 23 problems. Kurt Gödel (1940) and Paul Cohen (1963) proved it is independent of ZFC set theory: it can neither be proved nor disproved from the standard axioms. Whether to accept it as an axiom remains a philosophical question. <!-- LITERATURE-TRIAGE:BEGIN --> ## Li...	open	L5: Millennium Prize	5	Set Theory	null	Georg Cantor	1,878	1,234	67	train
NT-006	Legendre's Conjecture	For every positive integer $n$, there exists a prime number between $n^2$ and $(n+1)^2$.	This conjecture about the distribution of prime numbers was proposed by Adrien-Marie Legendre in 1808. Despite significant progress in prime number theory, including the prime number theorem, this simple statement remains unproven. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status:...	open	L3: Advanced	3	Number Theory	null	Adrien-Marie Legendre	1,808	432	26	train
NT-007	Are there infinitely many Mersenne primes?	Are there infinitely many prime numbers of the form $M_p = 2^p - 1$ where $p$ is prime?	Mersenne primes are primes of the form $2^p - 1$. As of 2024, only 51 Mersenne primes are known, with the largest being $2^{82,589,933} - 1$. It is conjectured that infinitely many exist, but this remains unproven. They are important for computational number theory and the GIMPS distributed computing project. <!-- LIT...	open	L4: Expert	4	Number Theory	null	null	null	654	38	train
NT-008	Are there infinitely many perfect powers in the Fibonacci sequence?	Besides 1, 8, and 144, are there any other perfect powers (numbers of the form $a^b$ where $a, b > 1$) in the Fibonacci sequence?	The Fibonacci sequence has only three known perfect powers: $F_1 = F_2 = 1 = 1^n$, $F_6 = 8 = 2^3$, and $F_{12} = 144 = 12^2$. It is conjectured that these are the only ones, but this remains unproven despite extensive computational searches. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) ...	open	L3: Advanced	3	Number Theory	null	null	null	345	21	train
NT-009	Gilbreath's Conjecture	Starting with the sequence of primes and repeatedly taking absolute differences of consecutive terms, the first term of each row is always 1.	Norman Gilbreath observed in 1958 that applying the forward difference operator to the sequence of primes appears to always yield 1 as the first element. Despite being verified computationally for the first $10^{13}$ primes, no proof exists. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) *...	open	L3: Advanced	3	Number Theory	null	Norman Gilbreath	1,958	287	15	train
COMB-003	Ramsey Number R(5,5)	What is the exact value of $R(5,5)$, the smallest number $n$ such that any 2-coloring of the edges of $K_n$ contains a monochromatic $K_5$?	Ramsey theory asks how large a structure must be to guarantee a certain property. The Ramsey number $R(5,5)$ is known to lie between 43 and 48, but the exact value remains unknown. As Joel Spencer said, "Erdős asks us to imagine an alien force, demanding the value of $R(5,5)$ or they will destroy our planet... our best...	open	L3: Advanced	3	Combinatorics	null	null	null	543	32	train
GT-003	The Graceful Tree Conjecture	Every tree can be gracefully labeled: vertices can be assigned distinct labels from $\{0, 1, \ldots, \|E\|\}$ such that edge labels (absolute differences) are all distinct.	The graceful labeling conjecture, proposed by Alexander Rosa in 1967, asks whether every tree admits a graceful labeling. It has been verified for many classes of trees including paths, caterpillars, and trees with at most 35 vertices, but remains open in general. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review ...	open	L3: Advanced	3	Graph Theory	null	Alexander Rosa	1,967	321	18	train
GEO-003	The Kakeya Conjecture	A Kakeya set (containing a unit line segment in every direction) in $\mathbb{R}^n$ must have Hausdorff dimension $n$.	The Kakeya conjecture concerns the minimal "size" of sets containing line segments in all directions. It has deep connections to harmonic analysis and PDE. The conjecture is known in dimension 2 but remains open for $n \geq 3$. It would have important implications for the restriction conjecture in Fourier analysis. <!...	open	L4: Expert	4	Geometry	null	null	null	432	24	train
GEO-004	The Moving Sofa Problem	What is the largest area of a shape that can be maneuvered through an L-shaped corridor of unit width?	This classic problem in geometric optimization asks for the largest "sofa" that can navigate a right-angled hallway. The best known lower bound is approximately 2.2195 (Gerver's sofa, 1992), and the upper bound is $2\sqrt{2} \approx 2.8284$. The exact answer remains unknown. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literat...	open	L3: Advanced	3	Geometry	null	null	null	567	41	train
TOP-002	The Volume Conjecture	For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement.	The volume conjecture, proposed by Rinat Kashaev in 1995 and generalized by Murakami and Murakami in 2001, connects quantum invariants of knots to their classical geometric properties. It relates quantum topology to hyperbolic geometry and has been verified for many knots but remains unproven in general. <!-- LITERATU...	open	L4: Expert	4	Topology	null	Rinat Kashaev	1,995	298	17	train
AG-001	The Standard Conjectures on Algebraic Cycles	A collection of conjectures about algebraic cycles on smooth projective varieties, including Lefschetz standard conjecture and Künneth standard conjecture.	The standard conjectures, formulated by Alexander Grothendieck in the 1960s, concern the theory of algebraic cycles and their cohomology. They would have profound consequences for algebraic geometry, including the independence of Betti numbers from the choice of Weil cohomology theory. The Hodge conjecture would follow...	open	L5: Millennium Prize	5	Algebraic Geometry	null	Alexander Grothendieck	1,965	432	23	train
AG-002	The Abundance Conjecture	For a minimal model $X$ of non-negative Kodaira dimension, the canonical divisor $K_X$ is semi-ample.	The abundance conjecture is a major open problem in birational algebraic geometry and the minimal model program. It predicts that canonical divisors on minimal models have good positivity properties. The conjecture is known in dimension 3 and in many special cases, but remains open in dimension 4 and higher. <!-- LITE...	open	L4: Expert	4	Algebraic Geometry	null	null	null	298	16	train
ALG-003	The Köthe Conjecture	A ring has no non-zero nil ideal (an ideal all of whose elements are nilpotent) if and only if it has no non-zero nil one-sided ideal.	The Köthe conjecture concerns the structure of rings with nilpotent elements. Proposed by Gottfried Köthe in 1930, it remains one of the oldest open problems in ring theory. Various special cases have been resolved, but the general conjecture remains open. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked...	open	L3: Advanced	3	Algebra	null	Gottfried Köthe	1,930	234	13	train
PDE-001	The Regularity Problem for Euler Equations	Do solutions to the 3D Euler equations for incompressible fluid flow remain smooth for all time, given smooth initial data?	The Euler equations describe the motion of inviscid (frictionless) fluids. While the Navier-Stokes equations include viscosity and are a Millennium Prize Problem, the regularity of Euler equations is also a major open question. Finite-time blowup would have profound implications for fluid dynamics. <!-- LITERATURE-TRI...	open	L4: Expert	4	Partial Differential Equations	null	null	null	456	26	train
SET-002	Singular Cardinals Hypothesis	If $\kappa$ is a singular strong limit cardinal, then $2^\kappa = \kappa^+$.	The singular cardinals hypothesis, formulated by Paul Erdős and András Hajnal, concerns the behavior of the power set operation on infinite cardinals. It sits between the generalized continuum hypothesis and ZFC. Its consistency and independence status remains a major open problem in set theory. <!-- LITERATURE-TRIAGE...	open	L4: Expert	4	Set Theory	null	Paul Erdős and András Hajnal	null	287	15	train
SET-003	Whitehead Problem	Is every abelian group $A$ such that $\text{Ext}^1(A, \mathbb{Z}) = 0$ a free abelian group?	The Whitehead problem, posed by J.H.C. Whitehead in 1950, asks about the structure of certain abelian groups. Shelah proved in 1973 that the problem is independent of ZFC: it is true under the constructible universe axiom (V=L) but can be false under other set-theoretic axioms. <!-- LITERATURE-TRIAGE:BEGIN --> ## Lite...	open	L4: Expert	4	Set Theory	null	J.H.C. Whitehead	1,950	198	11	train
CS-001	The Unique Games Conjecture	For certain constraint satisfaction problems (unique games), it is NP-hard to approximate the maximum fraction of satisfiable constraints beyond a certain threshold.	The Unique Games Conjecture, proposed by Subhash Khot in 2002, has become central to computational complexity theory. If true, it would imply optimal hardness results for many approximation problems. Khot was awarded the Nevanlinna Prize in 2014 for this work, despite the conjecture remaining unresolved. <!-- LITERATU...	open	L4: Expert	4	Computer Science	null	Subhash Khot	2,002	543	32	train
CS-002	The Polynomial Hirsch Conjecture	The diameter of the graph of a $d$-dimensional polytope with $n$ facets is bounded by a polynomial in $d$ and $n$.	The original Hirsch conjecture (diameter at most $n - d$) was disproved in 2010 by Francisco Santos. The polynomial Hirsch conjecture is a weaker version that remains open and is important for understanding the complexity of the simplex algorithm for linear programming. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature r...	open	L3: Advanced	3	Computer Science	null	null	null	321	18	train
HIL-012	Hilbert's 12th Problem: Extension of Kronecker-Weber Theorem	Extend the Kronecker-Weber theorem on abelian extensions of the rationals to any base number field.	Hilbert's 12th problem, posed in 1900, asks for an explicit construction of abelian extensions of number fields, generalizing the Kronecker-Weber theorem which states that every abelian extension of the rationals is contained in a cyclotomic field. Despite significant progress in class field theory, the problem of find...	open	L5: Millennium Prize	5	Number Theory	Hilbert's 23 Problems	David Hilbert	1,900	345	19	train
HIL-016	Hilbert's 16th Problem: Topology of Algebraic Curves and Limit Cycles	Determine the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$, and investigate the topology of real algebraic curves and surfaces.	Posed by David Hilbert in 1900, this two-part problem concerns (1) the topology of real algebraic varieties and (2) the limit cycles of planar polynomial differential equations. While it was shown in 1991-1992 by Ilyashenko and Écalle that polynomial vector fields have finitely many limit cycles, the question of whethe...	open	L5: Millennium Prize	5	Geometry	Hilbert's 23 Problems	David Hilbert	1,900	432	24	train
LAN-004	Landau's Fourth Problem: Primes of the Form n² + 1	Are there infinitely many primes of the form $n^2 + 1$?	One of Landau's four problems presented at the 1912 International Congress of Mathematicians, this asks whether there are infinitely many primes that are one more than a perfect square. Examples include 2, 5, 17, 37, 101, 197, 257, 401. Despite being simple to state, it has remained unsolved for over 110 years and is c...	open	L4: Expert	4	Number Theory	Landau's Problems	Edmund Landau	1,912	398	22	train
SMA-004	Smale's 4th Problem: Integer Zeros of Polynomials	Find efficient algorithms for deciding whether a polynomial with integer coefficients has an integer root.	Part of Stephen Smale's 18 problems for the 21st century (1998), this problem asks for polynomial-time algorithms to determine if a polynomial equation has integer solutions. This is related to Hilbert's 10th problem, which was shown to be undecidable in general, but specific cases and algorithms with better complexity...	open	L4: Expert	4	Computer Science	Smale's Problems	Stephen Smale	1,998	287	16	train
SMA-005	Smale's 5th Problem: Height Bounds for Diophantine Curves	Find effective uniform bounds for the heights of rational points on algebraic curves.	From Smale's 1998 list, this problem addresses the challenge of bounding the size of integer solutions to algebraic equations. While Faltings proved that curves of genus > 1 have finitely many rational points, the question of effective bounds on their heights remains a major open problem in arithmetic geometry. <!-- L...	open	L4: Expert	4	Algebraic Geometry	Smale's Problems	Stephen Smale	1,998	234	13	train
SMA-006	Smale's 6th Problem: Finiteness of Central Configurations	For the Newtonian $n$-body problem with positive masses, are there only finitely many central configurations (relative equilibria) for each $n$?	This problem from Smale's 1998 list concerns celestial mechanics and asks whether gravitating bodies can have only finitely many stable equilibrium configurations. The question is known to be true for n = 3 and n = 4, but remains open for n ≥ 5. It connects classical mechanics with algebraic geometry. <!-- LITERATURE-...	open	L4: Expert	4	Geometry	Smale's Problems	Stephen Smale	1,998	198	11	train
SMA-007	Smale's 7th Problem: Distribution of Points on the 2-Sphere	What is the optimal arrangement of $n$ points on the 2-sphere to minimize energy for various potential functions?	Smale's 7th problem (1998) asks for the configuration that minimizes various energy functionals for points on a sphere. This includes the Thomson problem (electrons on a sphere) and related optimization questions. Solutions are known for small n and highly symmetric cases, but the general problem remains open and conne...	open	L3: Advanced	3	Geometry	Smale's Problems	Stephen Smale	1,998	267	15	train
SMA-009	Smale's 9th Problem: Linear Programming in Polynomial Time	Find a strongly polynomial algorithm for linear programming.	Smale's 9th problem (1998) asks whether there exists an algorithm for linear programming whose running time is polynomial in the number of constraints and variables, independent of the bit-size of the input. While linear programming is solvable in polynomial time, no strongly polynomial algorithm is known for the gener...	open	L4: Expert	4	Computer Science	Smale's Problems	Stephen Smale	1,998	312	18	train
SMA-010	Smale's 10th Problem: The Pugh Closing Lemma	Is the $C^r$ closing lemma true for dynamical systems?	The closing lemma in dynamical systems theory asks whether, for a diffeomorphism with a nonwandering point, there is an arbitrarily small perturbation that makes that point periodic. Pugh proved a $C^1$ version in 1967, but the $C^r$ version for r ≥ 2 remains open. This is Smale's 10th problem from his 1998 list. <!--...	open	L4: Expert	4	Geometry	Smale's Problems	Stephen Smale	1,998	176	9	train
SMA-016	The Jacobian Conjecture	If $F: \mathbb{C}^n \to \mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible.	The Jacobian conjecture, proposed in 1939 and featured as Smale's 16th problem (1998), asks whether polynomial maps with nowhere-vanishing Jacobian determinant are necessarily invertible. Despite its elementary statement, it has resisted numerous attempts at proof. The conjecture is known to be true in dimension 1 and ...	open	L4: Expert	4	Algebra	Smale's Problems	Ott-Heinrich Keller	1,939	298	17	train
COMB-005	Frankl's Union-Closed Sets Conjecture	For every finite union-closed family of sets (other than the empty family), there exists an element that belongs to at least half of the sets.	Proposed by Péter Frankl in 1979, this is one of the best-known open problems in combinatorics. A union-closed family is a collection of sets closed under taking unions. Despite its simple statement, the conjecture has attracted many attempted proofs. Recent progress (2022-2024) has shown lower bounds: some element mus...	open	L3: Advanced	3	Combinatorics	null	Péter Frankl	1,979	389	21	train
GEO-005	Inscribed Square Problem (Toeplitz Conjecture)	Does every simple closed curve in the plane contain all four vertices of some square?	The inscribed square problem, also called the square peg problem or Toeplitz conjecture, was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve (simple closed curve) inscribes a square. The conjecture is known to be true for convex curves, piecewise smooth curves, and many special cases, but remains ope...	open	L4: Expert	4	Geometry	null	Otto Toeplitz	1,911	432	24	train
NT-010	Brocard's Problem	Find all integer solutions to $n! + 1 = m^2$.	Brocard's problem asks for all positive integers n such that n! + 1 is a perfect square. Only three solutions are known: (4, 5), (5, 11), and (7, 71), corresponding to 4! + 1 = 25, 5! + 1 = 121, and 7! + 1 = 5041. It has been verified computationally that no other solutions exist for n < 10^9, but it remains unproven w...	open	L3: Advanced	3	Number Theory	null	null	null	345	19	train
GT-004	The Cycle Double Cover Conjecture	Every bridgeless graph has a cycle double cover: a collection of cycles that covers each edge exactly twice.	The cycle double cover conjecture, proposed independently by Paul Seymour and Gábor Szekeres in the 1970s, is a major open problem in graph theory. It has been verified for many classes of graphs, including planar graphs and graphs with small genus. The conjecture is related to the snark conjecture and has connections ...	open	L4: Expert	4	Graph Theory	null	null	null	298	17	train
NT-012	The Erdős-Straus Conjecture	For every integer $n \geq 2$, the equation $\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ has a solution in positive integers x, y, z.	The Erdős-Straus conjecture concerns Egyptian fractions (sums of unit fractions). Paul Erdős and Ernst G. Straus conjectured in 1948 that 4/n can always be expressed as the sum of three unit fractions. The conjecture has been verified for all n up to 10^17 and is known to hold for various infinite families, but a gener...	open	L3: Advanced	3	Number Theory	null	Paul Erdős and Ernst G. Straus	1,948	367	20	train
HIL-006	Hilbert's 6th Problem: Axiomatization of Physics	Develop a mathematical framework that axiomatizes physics, particularly mechanics, thermodynamics, and probability theory.	Hilbert's 6th problem (1900) calls for treating physics with the same mathematical rigor as geometry. While progress has been made (quantum mechanics axiomatization by von Neumann, some progress in quantum field theory), a complete axiomatization remains elusive, especially for areas like thermodynamics and a unified "...	open	L5: Millennium Prize	5	Mathematical Physics	Hilbert's 23 Problems	David Hilbert	1,900	345	19	train
HIL-013	Hilbert's 13th Problem: Seventh Degree Equations	Prove that the general equation of the seventh degree cannot be solved using functions of only two variables.	Hilbert's 13th problem (1900) asks whether seventh-degree equations can be solved using continuous functions of two variables. Vladimir Arnold and Andrey Kolmogorov showed in 1957 that any continuous function can be represented using functions of two variables, which contradicts Hilbert's expectation. However, the prob...	open	L4: Expert	4	Algebra	Hilbert's 23 Problems	David Hilbert	1,900	287	16	train
SMA-012	Smale's 12th Problem: Centralizers of Diffeomorphisms	Determine the structure of centralizers of generic diffeomorphisms.	Smale's 12th problem (1998) concerns the algebraic structure of diffeomorphisms that commute with a given diffeomorphism. The centralizer of a dynamical system reveals its symmetries. Smale conjectured that for generic diffeomorphisms, the centralizer should be trivial or nearly trivial. <!-- LITERATURE-TRIAGE:BEGIN -...	open	L4: Expert	4	Geometry	Smale's Problems	Stephen Smale	1,998	176	9	train
DARPA-002	The Dynamics of Networks	Develop high-dimensional mathematics to model and predict behavior in large-scale distributed networks.	DARPA challenge 2 (2007) addresses the need for mathematical tools to understand massive networks like the internet, social networks, and biological networks. Traditional graph theory becomes inadequate at scale, requiring new mathematical frameworks for network dynamics. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature...	open	L4: Expert	4	Graph Theory	DARPA's 23 Mathematical Challenges	DARPA	2,007	389	21	train
DARPA-004	21st Century Fluids	Extend classical fluid dynamics to handle complex substances like foams, suspensions, gels, and liquid crystals.	DARPA challenge 4 (2007) recognizes that most real-world fluids don't behave like the classical fluids of Navier-Stokes equations. New mathematics is needed for complex fluids with microstructure, non-Newtonian behavior, and multiphase dynamics. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17...	open	L4: Expert	4	Partial Differential Equations	DARPA's 23 Mathematical Challenges	DARPA	2,007	345	19	train
DARPA-005	Biological Quantum Field Theory	Apply quantum and statistical field theory methods to model and potentially control pathogen evolution.	DARPA challenge 5 (2007) proposes using the mathematical machinery of quantum field theory—developed for particle physics—to understand biological evolution and epidemiology. This could provide new ways to predict and control disease evolution. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17)...	open	L5: Millennium Prize	5	Mathematical Physics	DARPA's 23 Mathematical Challenges	DARPA	2,007	267	15	train
DARPA-008	Beyond Convex Optimization	Determine whether algebraic geometry can systematically replace linear algebra in optimization.	DARPA challenge 8 (2007) asks whether the powerful tools of algebraic geometry can extend optimization beyond the convex case. Most practical optimization problems are non-convex, and algebraic geometry may provide the framework for solving them systematically. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (ch...	open	L4: Expert	4	Computer Science	DARPA's 23 Mathematical Challenges	DARPA	2,007	312	18	train
DARPA-012	Mathematics of Quantum Computing	Develop the mathematics required to control the quantum world for computation.	DARPA challenge 12 (2007) calls for mathematical foundations of quantum computing, including quantum algorithms, quantum entanglement, and quantum error correction. While quantum computers exist, the mathematical theory of what they can compute and how to program them remains underdeveloped. <!-- LITERATURE-TRIAGE:BEG...	open	L5: Millennium Prize	5	Computer Science	DARPA's 23 Mathematical Challenges	DARPA	2,007	543	32	train
DARPA-013	Game Theory at Scale	Create scalable mathematics for differential games, replacing traditional PDE approaches.	DARPA challenge 13 (2007) addresses the limitations of classical game theory and differential games when dealing with many players. New mathematical frameworks are needed for multi-agent systems, from autonomous vehicles to economic markets to military strategy. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (c...	open	L4: Expert	4	Computer Science	DARPA's 23 Mathematical Challenges	DARPA	2,007	289	16	train
DARPA-020	Computation at Scale	Develop asymptotics for systems with massive degrees of freedom.	DARPA challenge 20 (2007) addresses the mathematical challenges of understanding systems with enormous numbers of variables—from climate models to protein folding to materials science. Traditional approaches fail at extreme scales, requiring new asymptotic methods. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review...	open	L4: Expert	4	Computer Science	DARPA's 23 Mathematical Challenges	DARPA	2,007	276	15	train
DARPA-023	Fundamental Laws of Biology	Identify governing principles for biological systems, analogous to physical laws.	DARPA challenge 23 (2007) poses perhaps the deepest question: Do fundamental mathematical laws govern biology the way physics is governed by laws? This challenge requires solutions to multiple preceding challenges and asks whether biology can be made as mathematically rigorous as physics. <!-- LITERATURE-TRIAGE:BEGIN ...	open	L5: Millennium Prize	5	Mathematical Physics	DARPA's 23 Mathematical Challenges	DARPA	2,007	498	29	train
DARPA-006	Computational Duality	Use mathematical duality and geometry as foundations for developing novel computational algorithms.	DARPA challenge 6 (2007) explores whether duality principles from mathematics can lead to breakthrough algorithms. Dualities connect seemingly different mathematical structures and may reveal hidden computational efficiencies. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: part...	open	L4: Expert	4	Computer Science	DARPA's 23 Mathematical Challenges	DARPA	2,007	198	11	train
DARPA-009	Physical Consequences of Perelman's Proof	Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales.	DARPA challenge 9 (2007) asks how Grisha Perelman's breakthrough in understanding 3-dimensional geometry can inform materials science, from nanostructures to macro-scale fabrication. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved Classification: PARTIAL-P...	open	L4: Expert	4	Topology	DARPA's 23 Mathematical Challenges	DARPA	2,007	267	15	train
DARPA-010	Algorithmic Origami and Biology	Strengthen mathematical theory for isometric and rigid embedding relevant to protein folding.	DARPA challenge 10 (2007) connects origami mathematics to biology. Protein folding is like origami at molecular scales, and better mathematical theory could revolutionize drug design and protein engineering. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved **Cl...	open	L4: Expert	4	Geometry	DARPA's 23 Mathematical Challenges	DARPA	2,007	298	17	train
DARPA-011	Optimal Nanostructures	Develop mathematics for creating optimal symmetric structures through nanoscale self-assembly.	DARPA challenge 11 (2007) seeks mathematical principles for designing nanostructures that self-assemble optimally. This combines crystallography, optimization, and molecular dynamics. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved Classification: PARTIAL-...	open	L4: Expert	4	Geometry	DARPA's 23 Mathematical Challenges	DARPA	2,007	223	12	train
DARPA-015	The Geometry of Genome Space	Establish appropriate distance metrics on genome space incorporating biological utility.	DARPA challenge 15 (2007) asks for a mathematical geometry of genetics. How "far apart" are two genomes? The answer depends on biology, not just counting mutations, requiring new geometric frameworks. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved **Classific...	open	L4: Expert	4	Geometry	DARPA's 23 Mathematical Challenges	DARPA	2,007	245	14	train
DARPA-016	Symmetries and Action Principles for Biology	Extend understanding of symmetries and action principles in biology to include robustness, modularity, evolvability, and variability.	DARPA challenge 16 (2007) seeks to identify fundamental symmetry principles in biology analogous to those in physics. Why are biological systems robust yet evolvable? Are there variational principles governing life? <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solve...	open	L5: Millennium Prize	5	Mathematical Physics	DARPA's 23 Mathematical Challenges	DARPA	2,007	312	18	train
DARPA-017	Geometric Langlands and Quantum Physics	Connect the Langlands program to fundamental physics symmetries.	DARPA challenge 17 (2007) explores deep connections between number theory (Langlands program) and quantum field theory. This could unify disparate areas of mathematics and physics. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved Classification: PARTIAL-PRO...	open	L5: Millennium Prize	5	Mathematical Physics	DARPA's 23 Mathematical Challenges	DARPA	2,007	356	20	train
DARPA-018	Arithmetic Langlands, Topology, and Geometry	Explore homotopy theory's role in Langlands programs.	DARPA challenge 18 (2007) connects topology (homotopy theory) with the Langlands program in number theory. These connections could revolutionize both fields. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved Classification: PARTIAL-PROGRESS **Current litera...	open	L5: Millennium Prize	5	Topology	DARPA's 23 Mathematical Challenges	DARPA	2,007	289	16	train
HIL-007	Hilbert's 7th Problem: Transcendence of Certain Numbers	If $\alpha$ is algebraic and irrational, and $\beta$ is algebraic and irrational, is $\alpha^\beta$ transcendental?	Hilbert's 7th problem (1900) was largely solved by Gelfond and Schneider independently in 1934 (Gelfond-Schneider theorem). However, cases involving non-algebraic irrational exponents remain open. For example, whether $e^e$ or $\pi^\pi$ are transcendental is unknown. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature revi...	open	L4: Expert	4	Number Theory	Hilbert's 23 Problems	David Hilbert	1,900	321	18	train
HIL-009	Hilbert's 9th Problem: Reciprocity Laws	Generalize the reciprocity law of number theory to arbitrary number fields.	Hilbert's 9th problem (1900) asks for extensions of quadratic reciprocity to general number fields. Emil Artin made progress with Artin reciprocity law (1927), but complete understanding of reciprocity in all cases remains an active research area. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-...	open	L5: Millennium Prize	5	Number Theory	Hilbert's 23 Problems	David Hilbert	1,900	234	13	train
HIL-011	Hilbert's 11th Problem: Quadratic Forms over Algebraic Number Fields	Extend the theory of quadratic forms with algebraic numerical coefficients.	Hilbert's 11th problem (1900) concerns arithmetic of quadratic forms over number fields. Partial progress has been made through class field theory and the Hasse-Minkowski theorem, but general questions about representations remain open. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) **Stat...	open	L4: Expert	4	Number Theory	Hilbert's 23 Problems	David Hilbert	1,900	198	11	train
HIL-014	Hilbert's 14th Problem: Finite Generation of Rings	Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?	Hilbert's 14th problem (1900) was answered negatively by Nagata in 1958, who found counterexamples. However, the problem remains interesting for special cases, and understanding when finite generation holds is an active area. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: solve...	open	L4: Expert	4	Algebra	Hilbert's 23 Problems	David Hilbert	1,900	176	9	train
HIL-015	Hilbert's 15th Problem: Schubert's Enumerative Calculus	Rigorously justify Schubert's enumerative geometry.	Hilbert's 15th problem (1900) calls for making Schubert's 19th century enumerative geometry rigorous. While intersection theory and Schubert calculus have been developed (Chow rings, Gromov-Witten theory), some classical problems remain open and new questions arise. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature revie...	open	L4: Expert	4	Algebraic Geometry	Hilbert's 23 Problems	David Hilbert	1,900	267	15	train
HIL-017	Hilbert's 17th Problem: Expression of Definite Forms	Can every non-negative rational function be expressed as a sum of squares of rational functions?	Hilbert's 17th problem (1900) was solved affirmatively by Artin in 1927: every non-negative polynomial can be written as a sum of squares of rational functions. However, questions about minimal representations and related problems in real algebraic geometry remain active. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature...	open	L3: Advanced	3	Algebra	Hilbert's 23 Problems	David Hilbert	1,900	198	11	train
HIL-018	Hilbert's 18th Problem: Polyhedra and Space-Filling	Are there only finitely many essentially different space-filling convex polyhedra? Is there a polyhedron which tiles space but not in a lattice arrangement?	Hilbert's 18th problem (1900) has multiple parts. Non-lattice tilings (aperiodic tilings) were discovered by Heesch and others. The Kepler conjecture about sphere packing was proved by Hales. However, classification questions about space-filling polyhedra remain open. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature rev...	open	L3: Advanced	3	Geometry	Hilbert's 23 Problems	David Hilbert	1,900	289	16	train
GREEN-002	Restricted Sumset Problem	Let $A \subset \mathbb{Z}$ be a set of $n$ integers. Is there a subset $S \subset A$ of size $(\log n)^{100}$ such that $S \hat{+} S$ is disjoint from $A$?	Here $S \hat{+} S$ denotes the restricted sumset $\{s_1 + s_2 : s_1, s_2 \in S, s_1 \neq s_2\}$. Problems of this type are also at least 50 years old, being once again mentioned (and attributed to joint discussions of Erdős and Moser). It is known from very recent work of Sanders that there is always such an $S$ with $...	open	L2: Intermediate	2	Combinatorics	Ben Green's 100 Open Problems	Erdős and Moser	null	123	7	train
GREEN-005	Product-Free Sets in Finite Groups	Which finite groups have the smallest largest product-free sets?	Kedlaya (2003) showed that every finite group $G$ of order $n$ has a product-free subset of size $\gg n^{11/14}$, using the classification of finite simple groups. Understanding which groups achieve the minimum and improving bounds remains an open question connecting group theory and combinatorics. <!-- LITERATURE-TRI...	open	L2: Intermediate	2	Algebra	Ben Green's 100 Open Problems	Kedlaya	2,003	134	7	train
GREEN-007	Ulam's Sequence	Define Ulam's sequence $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \ldots$ where $u_1 = 1, u_2 = 2$, and $u_{n+1}$ is the smallest number uniquely expressible as $u_i + u_j$ for $i < j \leq n$. Does this sequence have positive density? Can one explain its curious Fourier properties?	Ulam's sequence exhibits mysterious quasi-periodic behavior in its Fourier transform. While it appears to have density around $0.07$, proving it has positive density remains open. The sequence's additive structure and apparent regularity in numerical experiments are not well understood theoretically. <!-- LITERATURE-T...	open	L1: Tractable	1	Number Theory	Ben Green's 100 Open Problems	Stanisław Ulam	null	187	11	train
GREEN-008	Almost Sum-Free Sets	Suppose that $A \subset [N]$ has no more than $\varepsilon N^2$ solutions to $x + y = z$. Can one remove $\varepsilon' N$ elements to leave a sum-free set, where $\varepsilon' \to 0$ as $\varepsilon \to 0$, with a reasonable bound?	It is known that one can remove $\varepsilon' N$ elements to obtain a sum-free set, but the quantitative dependence of $\varepsilon'$ on $\varepsilon$ is very poor. Finding explicit reasonable bounds would significantly improve our understanding of the structure of almost sum-free sets. <!-- LITERATURE-TRIAGE:BEGIN --...	open	L2: Intermediate	2	Combinatorics	Ben Green's 100 Open Problems	null	null	109	6	train
GREEN-006	Sum-Free Subsets of [N]^d	Fix an integer $d$. What is the largest sum-free subset of $[N]^d$?	This multi-dimensional generalization asks for the maximum size of a set in the $d$-dimensional grid with no solutions to $x + y = z$. Lepsveridze and Sun (2023) determined the constants $c_3, c_4, c_5$ and confirmed that the "slice example" is asymptotically optimal in these cases. <!-- LITERATURE-TRIAGE:BEGIN --> ##...	open	L1: Tractable	1	Combinatorics	Ben Green's 100 Open Problems	null	null	118	7	train
GREEN-009	Progressions in Subsets of Z/NZ	Is $r_5(N) \ll N(\log N)^{-c}$? Is $r_4(\mathbb{F}_5^n) \ll N^{1-c}$ where $N = 5^n$?	Here $r_k(N)$ denotes the maximum size of a subset of $\{1, \ldots, N\}$ with no $k$-term arithmetic progression. Kelley-Meka (2024) resolved the $k=3$ case. For $k \geq 5$, Leng-Sah-Sawhney (2024) proved bounds of shape $r_k(N) \ll Ne^{-(\log \log N)^{c_k}}$. Finding polynomial savings remains a central challenge in a...	open	L2: Intermediate	2	Combinatorics	Ben Green's 100 Open Problems	null	null	142	8	train
GREEN-010	Roth's Theorem with Random Common Differences	Let $S \subset \mathbb{N}$ be random. Under what conditions is Roth's theorem for progressions of length 3 true with common differences in $S$?	This asks when Roth's theorem holds if we restrict common differences to a random set. Briët and Castro-Silva (2023) advanced bounds for odd $k$. The problem explores how randomness interacts with additive structure. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solv...	open	L1: Tractable	1	Combinatorics	Ben Green's 100 Open Problems	null	null	126	7	train
GREEN-012	Tuples in Dense Sets	Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15}N^{10}$ tuples $(x_1, \ldots, x_5, y_1, \ldots, y_5) \in G^{10}$ such that $x_i + y_j \in A$ whenever $j \in \{i, i+1, i+2\}$?	This problem asks about higher-order additive structures in dense sets. Deng-Tidor-Zhao (2023) considered this problem and conjectured a negative answer, suggesting the exponent might not be optimal. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved **Classifica...	open	L2: Intermediate	2	Combinatorics	Ben Green's 100 Open Problems	null	null	108	6	train
GREEN-013	4-term APs in Fourier Uniform Sets	Suppose that $A \subset \mathbb{Z}/N\mathbb{Z}$ has density $\alpha$ and is Fourier uniform (all Fourier coefficients of $1_A - \alpha$ are $o(N)$). Does $A$ contain at least $\gg \alpha^{100}N^2$ 4-term arithmetic progressions?	Fourier uniformity means the set "looks random" from a Fourier perspective. The question asks if this forces many 4-APs. Deng-Tidor-Zhao (2023) conjectured a negative answer, suggesting Fourier uniformity alone may not suffice. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: par...	open	L2: Intermediate	2	Combinatorics	Ben Green's 100 Open Problems	null	null	115	7	train
GREEN-015	Lipschitz AP-Free Graphs	Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{Z}\} \subset \mathbb{Z}^2$ is free of 3-term progressions?	This asks whether a "smooth" (Lipschitz) function can have a graph avoiding arithmetic progressions. The Lipschitz condition prevents wildly oscillating behavior, making AP-avoidance more constrained. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: open Classification: OPE...	open	L1: Tractable	1	Combinatorics	Ben Green's 100 Open Problems	null	null	121	7	train
GREEN-016	Linear Equation x + 3y = 2z + 2w	What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?	This asks about sets avoiding a specific linear configuration. Understanding which linear equations are easier or harder to avoid is a fundamental question in additive combinatorics. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: open Classification: OPEN-TRIAGE **Curren...	open	L1: Tractable	1	Combinatorics	Ben Green's 100 Open Problems	null	null	98	5	train
GREEN-017	Progressions in F_3^n with Boolean Common Differences	Suppose that $A \subset \mathbb{F}_3^n$ is a set of density $\alpha$. Under what conditions on $\alpha$ is $A$ guaranteed to contain a 3-term progression with nonzero common difference in $\{0, 1\}^n$?	This constrains the progression to have Boolean-like common differences. Bhangale-Khot-Minzer (2023) showed sets avoiding such progressions have density $\ll_p (\log \log \log n)^{-c_p}$, using extraordinarily difficult techniques. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status:...	open	L2: Intermediate	2	Combinatorics	Ben Green's 100 Open Problems	null	null	104	6	train
GREEN-018	Corner Problem in Product Sets	Suppose $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Are there $\gg_\alpha \|G\|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$?	This is a "corner-type" problem in the group product setting. Dense sets should contain many axis-aligned corners. The problem connects additive combinatorics with group theory. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved Classification: PARTIAL-PROGRE...	open	L1: Tractable	1	Combinatorics	Ben Green's 100 Open Problems	null	null	110	6	train
GREEN-020	Multidimensional Szemerédi Theorem Bounds	Find reasonable bounds for instances of the multidimensional Szemerédi theorem.	Szemerédi's theorem extends to multiple dimensions (finding combinatorial lines in dense sets). Pohoata-Zakharov (2024) improved bounds for skew corners to $N^{5/4}$. Quantitative bounds remain a major challenge. <!-- LITERATURE-TRIAGE:BEGIN --> ## Literature review (checked 2026-08-17) Status: partially_solved ...	open	L2: Intermediate	2	Combinatorics	Ben Green's 100 Open Problems	null	null	127	7	train

End of preview. Expand in Data Studio

🧠 Unsolved Math — Clean

8,626 curated open research problems in mathematics and CS — including 122 Millennium Prize Problems — deduplicated, schema-flattened, and packaged as proper parquet configs with an eval-only benchmark view.

A reasoning frontier dataset: every problem here is actually unsolved or partially solved — ideal for honest capability probing instead of contaminated benchmarks.

Clean derivative of ulamai/UnsolvedMath (8,785 problems). License unchanged: CC-BY-4.0.

🧹 Quality Pipeline

Step	Removed	Reason
Raw input	8,785	Nested JSON
Duplicate titles	153	Same problem listed twice
Empty statements	6	No problem body
Final clean	8,626	Flattened schema, deterministic splits

📊 Composition

By status:

Status	Problems
Open	4,354
Partially solved	3,580
Solved	692

By difficulty:

Level	Problems
L1: Tractable	908
L2: Intermediate	516
L3: Advanced	6,143
L4: Expert	937
L5: Millennium Prize	122

📦 Configs

Config	Rows	Description
`problems` (default)	8,626	Full metadata: statement, background, status, difficulty, category, provenance
`open_problems_eval`	4,354	Open problems only — benchmark/eval view
`research_prompts`	8,626	Instruction-formatted: "analyze this open problem, discuss partial results and obstacles"

🎯 Usage

from datasets import load_dataset

ds = load_dataset("saidutta69/unsolved-math-clean", "problems", split="train")

# Honest eval: only truly open problems
ev = load_dataset("saidutta69/unsolved-math-clean", "open_problems_eval")

# Research-style SFT prompts
rp = load_dataset("saidutta69/unsolved-math-clean", "research_prompts")
print(rp["train"][0]["instruction"][:300])

⚠️ Notes

These are unsolved problems — there are no reference answers. Use for capability probing, calibration of hedging behavior, and research discussion; not for accuracy scoring.
LaTeX notation preserved verbatim.
Millennium-level problems are intentionally near-impossible; expect models to fail honestly.

📜 Citation

@misc{unsolved-math-clean,
  author = {Sai Dutta Abhishek Dash (clean); original by ulamai},
  title = {Unsolved Math — Clean},
  year = {2026},
  publisher = {Hugging Face},
  howpublished = {\url{https://huggingface.co/datasets/saidutta69/unsolved-math-clean}}
}

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