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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 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 ... | 5 | open | Stephen Cook | 1,971 | 15 | 1 | 1,523 | 89 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 1,
"name": "millennium_prize",
"display_name": "Millennium Prize Problems",
"description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.",
"slug": "millennium-prize",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
2 | 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... | 5 | open | Bernhard Riemann | 1,859 | 1 | 1 | 2,341 | 156 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 1,
"name": "millennium_prize",
"display_name": "Millennium Prize Problems",
"description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.",
"slug": "millennium-prize",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
3 | 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... | 5 | open | Yang Chen-Ning and Robert Mills | 1,954 | 16 | 1 | 1,234 | 78 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
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"id": 1,
"name": "millennium_prize",
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"description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.",
"slug": "millennium-prize",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
4 | 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 ... | 5 | open | Claude-Louis Navier and George Gabriel Stokes | 1,822 | 9 | 1 | 1,456 | 89 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 1,
"name": "millennium_prize",
"display_name": "Millennium Prize Problems",
"description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.",
"slug": "millennium-prize",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
5 | 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... | 5 | open | Bryan Birch and Peter Swinnerton-Dyer | 1,960 | 1 | 1 | 1,123 | 67 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 1,
"name": "millennium_prize",
"display_name": "Millennium Prize Problems",
"description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.",
"slug": "millennium-prize",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
6 | MPP-006 | Hodge Conjecture | On a projective non-singular algebraic variety over $\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles. | The Hodge conjecture is a major open problem in algebraic geometry. It seeks to relate the topology of a smooth complex projective variety to its algebraic structure, specifically asserting that certain topological cycles are actually algebraic.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17... | 5 | open | William Vallance Douglas Hodge | 1,950 | 5 | 1 | 987 | 54 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 1,
"name": "millennium_prize",
"display_name": "Millennium Prize Problems",
"description": "Seven problems selected by the Clay Mathematics Institute in 2000, each with a $1,000,000 prize for solution.",
"slug": "millennium-prize",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
8 | 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 ... | 3 | open | null | null | 1 | null | 543 | 34 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
9 | 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... | 4 | open | null | null | 1 | null | 892 | 67 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
11 | NT-003 | Twin Prime Conjecture | Are there infinitely many twin primes? Twin primes are pairs of primes that differ by 2, such as (3, 5), (5, 7), (11, 13), (17, 19), (29, 31). | The twin prime conjecture is one of the oldest unsolved problems in number theory. In 2013, Yitang Zhang proved that there are infinitely many pairs of primes that differ by at most 70 million. This bound has since been reduced to 246, but the gap of 2 remains unproven.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature r... | 4 | open | null | null | 1 | null | 1,234 | 89 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
12 | 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... | 4 | open | Christian Goldbach | 1,742 | 1 | null | 1,567 | 112 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
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"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
13 | 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... | 5 | open | Joseph Oesterlé and David Masser | 1,985 | 1 | null | 876 | 45 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
10 | 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... | 3 | open | null | null | 2 | null | 421 | 28 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
14 | 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... | 4 | open | Hugo Hadwiger | 1,943 | 3 | null | 654 | 38 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
15 | 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... | 3 | open | Stanisław Ulam | 1,942 | 3 | null | 432 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
18 | TOP-001 | Smooth 4-Dimensional Poincaré Conjecture | Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$? | The smooth Poincaré conjecture in dimension 4 is the only remaining case of the generalized Poincaré conjecture. It has been solved in all other dimensions: dimension 3 by Perelman, higher dimensions by Smale, Freedman, and others. The 4-dimensional case is particularly difficult due to exotic smooth structures.
<!-- ... | 5 | open | null | null | 7 | null | 789 | 42 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
19 | 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... | 4 | open | null | null | 6 | null | 456 | 27 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
20 | 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 ... | 4 | open | null | null | 4 | null | 543 | 29 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
21 | 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... | 4 | open | Irving Kaplansky | 1,940 | 4 | null | 321 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
22 | 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... | 5 | open | Georg Cantor | 1,878 | 10 | null | 1,234 | 67 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
25 | 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:**... | 3 | open | Adrien-Marie Legendre | 1,808 | 1 | null | 432 | 26 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
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"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
26 | 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... | 4 | open | null | null | 1 | null | 654 | 38 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
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"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
27 | 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)
... | 3 | open | null | null | 1 | null | 345 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
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"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
28 | 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)
*... | 3 | open | Norman Gilbreath | 1,958 | 1 | null | 287 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
29 | 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... | 3 | open | null | null | 2 | null | 543 | 32 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
30 | COMB-004 | The Lonely Runner Conjecture | For any $n$ runners on a circular track with distinct constant speeds, each runner is "lonely" (distance at least $1/n$ from all others) at some time. | This combinatorial conjecture, proposed by J.M. Wills in 1967, has been verified for up to 7 runners but remains open for 8 or more. It has connections to Diophantine approximation and view-obstruction problems.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved
... | 3 | open | J.M. Wills | 1,967 | 2 | null | 234 | 14 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
31 | 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 ... | 3 | open | Alexander Rosa | 1,967 | 3 | null | 321 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
32 | 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.
<!... | 4 | open | null | null | 6 | null | 432 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
33 | 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... | 3 | open | null | null | 6 | null | 567 | 41 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
34 | 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... | 4 | open | Rinat Kashaev | 1,995 | 7 | null | 298 | 17 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 7,
"name": "topology",
"display_name": "Topology",
"description": "Properties preserved under continuous deformations.",
"slug": "topology",
"order_index": 7,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
36 | 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... | 5 | open | Alexander Grothendieck | 1,965 | 5 | null | 432 | 23 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
"display_name": "Algebraic Geometry",
"description": "Geometric objects defined by polynomial equations.",
"slug": "algebraic-geometry",
"order_index": 5,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
37 | 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... | 4 | open | null | null | 5 | null | 298 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 5,
"name": "algebraic_geometry",
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"description": "Geometric objects defined by polynomial equations.",
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"created_at": "2026-07-31T15:26:25.671Z"
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"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
38 | 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... | 3 | open | Gottfried Köthe | 1,930 | 4 | null | 234 | 13 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
40 | 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... | 4 | open | null | null | 9 | null | 456 | 26 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
41 | 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... | 4 | open | Paul Erdős and András Hajnal | null | 10 | null | 287 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
42 | 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... | 4 | open | J.H.C. Whitehead | 1,950 | 10 | null | 198 | 11 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 10,
"name": "set_theory",
"display_name": "Set Theory",
"description": "Foundations of mathematics, infinite sets, and cardinality.",
"slug": "set-theory",
"order_index": 10,
"created_at": "2026-07-31T15:26:25.671Z"
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"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
43 | 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... | 4 | open | Subhash Khot | 2,002 | 15 | null | 543 | 32 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
44 | 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... | 3 | open | null | null | 15 | null | 321 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
45 | 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... | 5 | open | David Hilbert | 1,900 | 1 | 2 | 345 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 2,
"name": "hilbert_problems",
"display_name": "Hilbert's 23 Problems",
"description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.",
"slug": "hilbert-problems",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
... | null | null | null | null | null | null | null | null | null | null | null | null | null |
46 | 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... | 5 | open | David Hilbert | 1,900 | 6 | 2 | 432 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 2,
"name": "hilbert_problems",
"display_name": "Hilbert's 23 Problems",
"description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.",
"slug": "hilbert-problems",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
... | null | null | null | null | null | null | null | null | null | null | null | null | null |
47 | 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... | 4 | open | Edmund Landau | 1,912 | 1 | 6 | 398 | 22 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | {
"id": 6,
"name": "landau_problems",
"display_name": "Landau's Problems",
"description": "Four basic problems about prime numbers posed by Edmund Landau at the 1912 International Congress of Mathematicians.",
"slug": "landau-problems",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
48 | 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... | 4 | open | Stephen Smale | 1,998 | 15 | 5 | 287 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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} | null | null | null | null | null | null | null | null | null | null | null | null | null |
49 | 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.
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} | null | null | null | null | null | null | null | null | null | null | null | null | null |
50 | 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.
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51 | 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... | 3 | open | Stephen Smale | 1,998 | 6 | 5 | 267 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 6,
"name": "geometry",
"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
"slug": "geometry",
"order_index": 6,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"level": 3,
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} | null | null | null | null | null | null | null | null | null | null | null | null | null |
52 | 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... | 4 | open | Stephen Smale | 1,998 | 15 | 5 | 312 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
53 | 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.
<!--... | 4 | open | Stephen Smale | 1,998 | 6 | 5 | 176 | 9 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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54 | 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 ... | 4 | open | Ott-Heinrich Keller | 1,939 | 4 | 5 | 298 | 17 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
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"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
55 | 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... | 3 | open | Péter Frankl | 1,979 | 2 | null | 389 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
56 | 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... | 4 | open | Otto Toeplitz | 1,911 | 6 | null | 432 | 24 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
57 | 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... | 3 | open | null | null | 1 | null | 345 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 3,
"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
59 | 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 ... | 4 | open | null | null | 3 | null | 298 | 17 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
60 | 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... | 3 | open | Paul Erdős and Ernst G. Straus | 1,948 | 1 | null | 367 | 20 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
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"description": "Properties of integers, prime numbers, Diophantine equations.",
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"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
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"level": 3,
"name": "L3: Advanced",
"description": "Difficult problems requiring specialized knowledge and sophisticated techniques.",
"color_class": "text-yellow-600 bg-yellow-50 border-yellow-200"
} | null | null | null | null | null | null | null | null | null | null | null | null | null | null |
61 | 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 "... | 5 | open | David Hilbert | 1,900 | 16 | 2 | 345 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 2,
"name": "hilbert_problems",
"display_name": "Hilbert's 23 Problems",
"description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.",
"slug": "hilbert-problems",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
... | null | null | null | null | null | null | null | null | null | null | null | null | null |
62 | 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... | 4 | open | David Hilbert | 1,900 | 4 | 2 | 287 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 4,
"name": "algebra",
"display_name": "Algebra",
"description": "Group theory, ring theory, field theory, and algebraic structures.",
"slug": "algebra",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
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"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | {
"id": 2,
"name": "hilbert_problems",
"display_name": "Hilbert's 23 Problems",
"description": "David Hilbert's list of 23 unsolved problems presented at the International Congress of Mathematicians in Paris in 1900.",
"slug": "hilbert-problems",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
... | null | null | null | null | null | null | null | null | null | null | null | null | null |
64 | 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 -... | 4 | open | Stephen Smale | 1,998 | 6 | 5 | 176 | 9 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
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"display_name": "Geometry",
"description": "Euclidean and non-Euclidean geometry, geometric structures.",
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} | null | null | null | null | null | null | null | null | null | null | null | null | null |
66 | 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... | 4 | open | DARPA | 2,007 | 3 | 4 | 389 | 21 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 3,
"name": "graph_theory",
"display_name": "Graph Theory",
"description": "Problems involving graphs, networks, and their properties.",
"slug": "graph-theory",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
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"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | {
"id": 4,
"name": "darpa_challenges",
"display_name": "DARPA's 23 Mathematical Challenges",
"description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.",
"slug": "darpa-challenges",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
68 | 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... | 4 | open | DARPA | 2,007 | 9 | 4 | 345 | 19 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 9,
"name": "pde",
"display_name": "Partial Differential Equations",
"description": "PDEs and their applications in physics and geometry.",
"slug": "pde",
"order_index": 9,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | {
"id": 4,
"name": "darpa_challenges",
"display_name": "DARPA's 23 Mathematical Challenges",
"description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.",
"slug": "darpa-challenges",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
69 | 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)... | 5 | open | DARPA | 2,007 | 16 | 4 | 267 | 15 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 16,
"name": "physics",
"display_name": "Mathematical Physics",
"description": "Problems at the intersection of mathematics and physics.",
"slug": "physics",
"order_index": 16,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 4,
"name": "darpa_challenges",
"display_name": "DARPA's 23 Mathematical Challenges",
"description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.",
"slug": "darpa-challenges",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
70 | 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... | 4 | open | DARPA | 2,007 | 15 | 4 | 312 | 18 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | {
"id": 4,
"name": "darpa_challenges",
"display_name": "DARPA's 23 Mathematical Challenges",
"description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.",
"slug": "darpa-challenges",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
71 | 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... | 5 | open | DARPA | 2,007 | 15 | 4 | 543 | 32 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 5,
"level": 5,
"name": "L5: Millennium Prize",
"description": "Millennium Prize Problems and problems of equivalent difficulty.",
"color_class": "text-purple-600 bg-purple-50 border-purple-200"
} | {
"id": 4,
"name": "darpa_challenges",
"display_name": "DARPA's 23 Mathematical Challenges",
"description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.",
"slug": "darpa-challenges",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
72 | 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... | 4 | open | DARPA | 2,007 | 15 | 4 | 289 | 16 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 15,
"name": "computer_science",
"display_name": "Computer Science",
"description": "Computational complexity, algorithms, and theoretical CS.",
"slug": "computer-science",
"order_index": 15,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 4,
"level": 4,
"name": "L4: Expert",
"description": "Very challenging problems at the frontier of mathematical research.",
"color_class": "text-red-600 bg-red-50 border-red-200"
} | {
"id": 4,
"name": "darpa_challenges",
"display_name": "DARPA's 23 Mathematical Challenges",
"description": "Mathematical challenges identified by DARPA to drive fundamental research in mathematics.",
"slug": "darpa-challenges",
"order_index": 4,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
End of preview. Expand in Data Studio
UnsolvedMath Dataset
✅ Paper: Open Mathematical Problems as an AI Reasoning Benchmark
A comprehensive curated collection of 15,458 open and partially solved mathematics problems across all domains and difficulty levels, including the largest collection of Erdős problems available in machine-readable format. Available for browsing at unsolvedmath.com.
Dataset Description
UnsolvedMath is a comprehensive dataset of unsolved and historically significant mathematical problems, organized by domain, difficulty level, and problem set. This dataset aggregates problems from prestigious collections including:
- Millennium Prize Problems
- Hilbert's 23 Problems
- Smale's 18 Problems for the 21st Century
- DARPA's 23 Mathematical Challenges
- Ben Green's 100 Open Problems
- Erdős Problems
- Kourovka Notebook New Problems
- Kirby's Problems in Low-Dimensional Topology
- OpenGarden / Open Problem Garden
- AMR Open Problem Lists
- AIM Workshop Problem Lists
- Oberwolfach Reports workshop problems and discussions
Dataset Summary
- Total Problems: 15458
- Erdős Problems: 632 problems with citations and references
- Version: 1.6.0
- Categories: 17 mathematical domains
- Difficulty Levels: 5 (L1: Tractable → L5: Millennium Prize)
- Problem Sets: 15 curated collections
- Format: JSON
- License: CC BY 4.0
New in v1.2.0
This release adds problems collected from public source lists in the AMR index. AIM workshop lists are added in v1.4.0 below.
- AMR Open Problem Lists: 3342 problems
Every AMR record retains its source URL, extraction method, and status/rights review notes in the background field.
New in v1.3.0
This release adds a research status audit for every AMR open-problem record. Each of the 3,342 AMR problems was investigated by an AI research fleet (literature triage with web-verified citations, solution attempts, and partial progress), and every report was then re-checked in a supervised verification pass (statement alignment, citation-fabrication screening, classification normalization, difficulty assignment). Each AMR problem now carries: a research_classification (SOLVED-BY-YOU, SOLVED-IN-LITERATURE, PARTIAL-PROGRESS, OPEN-TRIAGE), an updated status derived from the classification, an optional research_difficulty_suggested, and a research_summary. AMR difficulty levels are now differentiated across L2–L5 (previously all L3). The structured per-problem research notes (problem, literature status, work done, result, what remains) are provided in research_results.json. The v1.3 tag also preserves the individual Markdown reports in research/.
- Research audit: 3,342 AMR problems classified; 183 solved (181 in the literature, 2 by the AI fleet), 963 partial progress, 2,196 open after triage.
- New file:
research_results.json— structured per-problem research notes. - v1.3 archive directory:
research/— full per-problem research reports in Markdown, preserved under the v1.3 tag. - New fields per AMR problem:
research_classification,research_summary,research_difficulty_suggested.
Caveat: classifications and summaries are machine-generated research aids, not peer-reviewed results; SOLVED-* entries were citation-checked, but independent verification is recommended before citing.
New in v1.4.0
This release adds the complete canonical AIM Workshop Problem Lists corpus:
3,359 problems from 26 AIM domain files. Every problem uses its exact
AIM-... canonical identifier, preserves source/workshop provenance, and is
paired with the effective validated research attempt produced by the AIM
multi-agent research run. All problems added in v1.4 come from AIM workshops,
and each received one solution attempt using GPT-5.6 Sol at xhigh
reasoning effort. The structured reports include 351 new AI results: 174
full solutions and 177 counterexamples. They are machine-generated claims and
have not been peer reviewed.
- AIM research audit: 3,359 validated reports: 174 full solutions, 177 counterexamples, 2,589 partial results, 45 conditional results, 182 reductions, 150 context-only reports, 41 invalid-statement reports, and 1 heuristic result.
- Research classifications: 461
SOLVED-IN-LITERATURE, 43SOLVED-BY-YOU, 2,664PARTIAL-PROGRESS, and 191OPEN-TRIAGE. These classifications describe the status of the underlying problem and are not a novelty filter for the AI result. - Identifiers: exact
AIM-<DOMAIN>-<NUMBER>tags inproblem_numberandresearch_results.json. - Difficulty: conservative AMR-style assignment (L3 default, L4 for clearly live conjectural/frontier cases, L2 for context-only or invalid statements; no automatic L5 assignments).
This release also replaces the coarse AIM wording_corrected heuristic with
an individual statement-recovery audit for all 3,359 canonical AIM IDs. The
exact canonical original_statement is stored separately from the reviewed
clean_statement; unrecoverable or unsafe reconstructions remain null rather
than being silently promoted into the public problem text.
exact: 2,886corrected_verified: 61reconstructed_unverified: 385unrecoverable: 27
For exact and corrected_verified, the public statement uses the clean
formulation. For reconstructed_unverified and unrecoverable, it retains a
visible rendering of the canonical source instead of silently adopting a
conjectural repair. Full evidence is available in aim_statement_audit.json.
Caveat: AIM reports and novelty labels are machine-generated research aids, not peer-reviewed claims. Full-solution and counterexample labels require independent expert verification before citation.
New in v1.5.0
This release adds a dated literature triage to all 2,084 problem records
outside the AMR and AIM collections. No problems were added or removed, and
the existing AMR and AIM problem records and research reports are unchanged.
Each review was checked on 2026-08-17 and is appended to the corresponding
problem's background field, with a literature-status assessment, verified
partial progress, resolution check, remaining work, and source links.
- Literature status: 670 open, 1,215 partially solved, and 199 solved.
- Scope: only non-AMR and non-AIM records; 3,342 AMR and 3,359 AIM records are preserved from v1.4.
- Data change:
backgroundonly; statements, identifiers, categories, difficulty assignments, and existing status fields are unchanged.
The v1.5 literature labels are dated machine-generated research aids. They
supplement the legacy status field and require independent verification
before citation. A refutation or counterexample is classified as solved;
uncertain literature status is classified as partially solved.
New in v1.6.0
This release adds 6,673 open and partially solved problems from Oberwolfach Reports, spanning Volumes 1–23 (2004–2026) and 1,106 source reports. The records capture explicit problems, conjectures, and research questions from conference reports together with the mathematical discussion needed to understand their current status.
- New OWR problems: 6,673 (4,493 open; 2,180 partially solved).
- Provenance: every record retains its OWR identifier, source-report DOI, report citation, and source links used by the literature assessment.
- Literature triage: assessments were checked on 2026-08-21 or 2026-08-22.
- Difficulty: all imported OWR records receive the conservative L3 default; this is a corpus-level default rather than an expert ranking of each problem.
- Categories: OWR domains are assigned conservatively from workshop titles and statement terminology; ambiguous records remain Miscellaneous.
- Quality check: a fixed-seed blind sample of ten records found eight clear, mathematically identifiable problems and two weaker programmatic or malformed extractions. The latter remain in the inclusion-first corpus and are flagged by their own literature assessments. Users should consult the source report where exact formulation matters.
The OWR literature assessments are dated curation aids, not peer review. Status and source claims should be independently verified before citation.
Supported Tasks
- Mathematical research and exploration
- Mathematical question answering
- LaTeX/mathematical notation processing
- Problem classification and organization
- Educational content generation
Dataset Structure
Data Files
The dataset consists of multiple JSON files:
- problems.json - Main dataset containing all problems
- categories.json - Mathematical domain classifications
- difficulty_levels.json - 5-tier difficulty system
- sets.json - Problem set metadata (Millennium Prize, Hilbert's 23, etc.)
- dataset.json - Combined file with all data
- statistics.json - Dataset statistics
- research_results.json - Per-problem AMR and AIM research notes (status, literature, result, what remains)
- aim_statement_audit.json - Per-problem AIM statement-recovery evidence and verification status
OWR records additionally expose source_citation, literature_assessment, literature_sources, and literature_checked_at.
Data Fields
Problems
Each problem contains:
id(int): Unique identifiertitle(string): Problem titlestatement(string): Complete problem statement with LaTeX notationbackground(string, optional): Historical context and backgroundcategory(object): Mathematical domainid,name,display_name,description,slug
difficulty(object): Difficulty classificationid,level,name,description,color_class
status(string): "open", "solved", or "partially_solved"source_url(string, optional): Reference URLsets(array, optional): Associated problem setstags(array, optional): Additional tagsyear_proposed(int, optional): Year the problem was first posedsolved_year(int, optional): Year solved (if applicable)solved_by(string, optional): Solver's nameprize_amount(int, optional): Prize money (USD)created_at(string): Timestampresearch_classification(string, optional): research verdict, e.g.SOLVED-IN-LITERATURE,PARTIAL-PROGRESS,OPEN-TRIAGEresearch_summary(string, optional): 2–3 paragraph summary of the research findingsresearch_difficulty_suggested(string, optional): suggested difficulty level when it differs from the default
Categories
17 mathematical domains:
- Number Theory: Properties of integers, prime numbers, Diophantine equations.
- Combinatorics: Counting problems, graph theory, discrete structures.
- Graph Theory: Problems involving graphs, networks, and their properties.
- Algebra: Group theory, ring theory, field theory, and algebraic structures.
- Algebraic Geometry: Geometric objects defined by polynomial equations.
- Geometry: Euclidean and non-Euclidean geometry, geometric structures.
- Topology: Properties preserved under continuous deformations.
- Analysis: Limits, continuity, calculus, and function theory.
- Partial Differential Equations: PDEs and their applications in physics and geometry.
- Set Theory: Foundations of mathematics, infinite sets, and cardinality.
- Dynamical Systems: Problems about long-term behavior of deterministic systems, Hamiltonian dynamics, and periodic orbits.
- Computer Science: Computational complexity, algorithms, and theoretical CS.
- Mathematical Physics: Problems at the intersection of mathematics and physics.
- Group Theory: Problems about groups, group actions, representations, and related algebraic structures.
- Logic: Problems in mathematical logic, model theory, proof theory, and finite model theory.
- Probability: Problems involving probability theory, stochastic processes, and random structures.
- Miscellaneous: Problems whose source classification does not fit the main mathematical categories.
Difficulty Levels
- L1: Tractable: Problems that may be within reach with current techniques. Reserved for future additions.
- L2: Intermediate: Challenging problems requiring solid mathematical background. Reserved for future additions.
- L3: Advanced: Difficult problems requiring specialized knowledge and sophisticated techniques.
- L4: Expert: Very challenging problems at the frontier of mathematical research.
- L5: Millennium Prize: Millennium Prize Problems and problems of equivalent difficulty.
Dataset Statistics
Problems by Difficulty
- L1: Tractable: 916
- L2: Intermediate: 523
- L3: Advanced: 12919
- L4: Expert: 963
- L5: Millennium Prize: 137
Problems by Category
- Algebra: 927
- Algebraic Geometry: 830
- Analysis: 1143
- Combinatorics: 744
- Computer Science: 642
- Dynamical Systems: 667
- Geometry: 1967
- Graph Theory: 1222
- Group Theory: 848
- Logic: 286
- Mathematical Physics: 352
- Miscellaneous: 1530
- Number Theory: 1268
- Partial Differential Equations: 270
- Probability: 798
- Set Theory: 77
- Topology: 1887
Problems by Status
- Open: 8955
- Partially Solved: 5807
- Solved: 696
Usage
Loading the Dataset
from datasets import load_dataset
# Load the full dataset
dataset = load_dataset("ulamai/UnsolvedMath", data_files="problems.json")
# Or load individual files
problems = load_dataset("ulamai/UnsolvedMath", data_files="problems.json")
categories = load_dataset("ulamai/UnsolvedMath", data_files="categories.json")
Example: Filtering by Difficulty
import json
with open('problems.json', 'r') as f:
problems = json.load(f)
# Get all Millennium Prize problems (L5)
millennium_problems = [
p for p in problems
if p.get('difficulty', {}).get('level') == 5
]
print(f"Found {len(millennium_problems)} Millennium Prize problems")
Example: LaTeX Rendering
# Problems contain LaTeX notation in the statement field
problem = problems[0]
print(problem['statement'])
# Use a LaTeX renderer like matplotlib or sympy to display
from sympy import latex, sympify
# ... render LaTeX content
Data Collection and Curation
This dataset was curated from:
- Official Millennium Prize Problems documentation
- Historical mathematical problem collections
- Published research papers and mathematical surveys
- Reputable mathematical organizations (Clay Mathematics Institute, AMS, etc.)
All problems include:
- Accurate mathematical statements with LaTeX notation
- Historical context and background
- Proper attribution and source references
- Classification by domain and difficulty
Ethical Considerations
- Academic Integrity: This dataset is for research and educational purposes
- Attribution: All problems are properly attributed to their original sources
- Open Problems: Status accuracy maintained to the best of our knowledge as of the dataset creation date
- Updates: Some problems may be solved after dataset publication
Limitations
- The dataset represents a curated selection, not an exhaustive list of all unsolved problems
- Problem difficulty is subjective and based on expert consensus
- LaTeX notation may require preprocessing for some applications
- Status (open/solved) should be verified for time-sensitive applications
- Some AMR and AIM records retain
NEEDS_REVIEWstatus or rights notes from the source audit; consult each record's provenance before reuse - Some OWR records are broad research directions or imperfect extractions; their literature assessments flag known formulation limitations
Citation
If you use this dataset in your research, please cite:
@misc{unsolvedmath2026,
title={UnsolvedMath: A Curated Collection of Open Mathematics Problems},
author={UnsolvedMath Contributors},
year={2026},
howpublished={\url{https://huggingface.co/datasets/ulamai/UnsolvedMath}},
}
Additional Information
Dataset Curators
UnsolvedMath project contributors
Licensing Information
The UnsolvedMath curation and original metadata are released under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Linked source material remains subject to its own terms; recent open-access Oberwolfach Reports identify CC BY-SA 4.0 on their article pages, and those source terms are not superseded by this dataset card.
You are free to:
- Share — copy and redistribute the material in any medium or format
- Adapt — remix, transform, and build upon the material for any purpose, even commercially
Under the following terms:
- Attribution — You must give appropriate credit and indicate if changes were made
Contact
For questions, issues, or contributions:
- Website: unsolvedmath.com
- Dataset: huggingface.co/datasets/ulamai/UnsolvedMath
Generated: 2026-08-25T00:00:00Z Version: 1.6.0
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