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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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"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,
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"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"
} | {
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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 |
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"
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"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"
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"level": 4,
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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"
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"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 |
74 | 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... | 4 | open | DARPA | 2,007 | 15 | 4 | 276 | 15 | 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,
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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"
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"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 |
75 | 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 ... | 5 | open | DARPA | 2,007 | 16 | 4 | 498 | 29 | 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",
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"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 |
76 | 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... | 4 | open | DARPA | 2,007 | 15 | 4 | 198 | 11 | 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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"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 |
77 | DARPA-007 | Occam's Razor in Many Dimensions | Find lower bounds for sensing complexity as data collection grows, addressing entropy maximization. | DARPA challenge 7 (2007) asks for mathematical principles governing data compression and sensing in high dimensions. As sensors become ubiquitous, we need mathematical theory for how much data is truly necessary.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** partially_solved ... | 4 | open | DARPA | 2,007 | 15 | 4 | 234 | 13 | 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 |
78 | 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... | 4 | open | DARPA | 2,007 | 7 | 4 | 267 | 15 | 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"
} | {
"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 |
79 | 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... | 4 | open | DARPA | 2,007 | 6 | 4 | 298 | 17 | 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"
} | {
"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 |
80 | 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-... | 4 | open | DARPA | 2,007 | 6 | 4 | 223 | 12 | 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"
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"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 |
81 | 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... | 4 | open | DARPA | 2,007 | 6 | 4 | 245 | 14 | 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"
} | {
"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 |
82 | 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... | 5 | open | DARPA | 2,007 | 16 | 4 | 312 | 18 | 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 |
83 | 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... | 5 | open | DARPA | 2,007 | 16 | 4 | 356 | 20 | 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",
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"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 |
84 | 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... | 5 | open | DARPA | 2,007 | 7 | 4 | 289 | 16 | 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"
} | {
"id": 4,
"name": "darpa_challenges",
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"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 |
85 | 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... | 4 | open | David Hilbert | 1,900 | 1 | 2 | 321 | 18 | 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": 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 |
86 | 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-... | 5 | open | David Hilbert | 1,900 | 1 | 2 | 234 | 13 | 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 |
87 | 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... | 4 | open | David Hilbert | 1,900 | 1 | 2 | 198 | 11 | 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": 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 |
88 | 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... | 4 | open | David Hilbert | 1,900 | 4 | 2 | 176 | 9 | 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"
} | {
"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 |
89 | 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... | 4 | open | David Hilbert | 1,900 | 5 | 2 | 267 | 15 | 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": 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": 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 |
90 | 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... | 3 | open | David Hilbert | 1,900 | 4 | 2 | 198 | 11 | 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"
} | {
"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 |
91 | 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... | 3 | open | David Hilbert | 1,900 | 6 | 2 | 289 | 16 | 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"
} | {
"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 |
92 | GREEN-001 | Large Sum-Free Sets | Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $n/3 + \Omega(n)$, where $\Omega(n) \to \infty$ as $n \to \infty$? | This is a pretty old and increasingly notorious problem, first mentioned over 50 years ago. The best known bounds are in Bourgain's paper, where he shows that there is necessarily a sum-free set of size at least $(n+2)/3$. In fact, Eberhard, Manners and Green (unpublished) worked out a proof that Problem 1 has a positi... | 2 | open | Erdős and Cameron | null | 2 | 3 | 145 | 8 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
93 | 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 $... | 2 | open | Erdős and Moser | null | 2 | 3 | 123 | 7 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
95 | 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... | 2 | open | Kedlaya | 2,003 | 4 | 3 | 134 | 7 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
96 | 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... | 1 | open | Stanisław Ulam | null | 1 | 3 | 187 | 11 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
97 | 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 --... | 2 | open | null | null | 2 | 3 | 109 | 6 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
98 | 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 -->
##... | 1 | open | null | null | 2 | 3 | 118 | 7 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
99 | 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... | 2 | open | null | null | 2 | 3 | 142 | 8 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
100 | 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... | 1 | open | null | null | 2 | 3 | 126 | 7 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
102 | 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... | 2 | open | null | null | 2 | 3 | 108 | 6 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
103 | 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... | 2 | open | null | null | 2 | 3 | 115 | 7 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
104 | 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... | 1 | open | null | null | 2 | 3 | 121 | 7 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
105 | 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... | 1 | open | null | null | 2 | 3 | 98 | 5 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
106 | 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:**... | 2 | open | null | null | 2 | 3 | 104 | 6 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
107 | 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... | 1 | open | null | null | 2 | 3 | 110 | 6 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
108 | 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 ... | 2 | open | null | null | 2 | 3 | 127 | 7 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
109 | GREEN-021 | Large Sieve and Quadratic Sets | Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic? | Sieve methods remove arithmetic structure from sets. This problem asks whether a set that "survives" a large sieve and has size $\sim N^2$ must actually be a quadratic sequence or similar structured set. Understanding the structure of sieved sets is fundamental in analytic number theory.
<!-- LITERATURE-TRIAGE:BEGIN -... | 1 | open | null | null | 1 | 3 | 87 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
110 | GREEN-022 | Small Sieve Maximal Sets | Suppose that a small sieve process leaves a set of maximal size. What is the structure of that set? | When a small sieve (sieving by small primes) leaves the maximum possible density of survivors, what structure must the original set have? This connects sieve theory with the structural theory of sets in number theory.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** open
**Clas... | 1 | open | null | null | 1 | 3 | 82 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 1,
"name": "number_theory",
"display_name": "Number Theory",
"description": "Properties of integers, prime numbers, Diophantine equations.",
"slug": "number-theory",
"order_index": 1,
"created_at": "2026-07-31T15:26:25.670Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
111 | GREEN-023 | Large Cosets in Iterated Sumsets | Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\log(1/\alpha))$? | This asks how many times we must add a set to itself before it contains a large subspace coset. Kosciuszko (2024), building on Konyagin, showed that $mA - mA$ contains a subspace of dimension $\geq n - O(\log^{3+\eta}(1/\alpha))$ for suitable $m$. The problem asks if fewer iterations suffice.
<!-- LITERATURE-TRIAGE:BE... | 2 | open | null | null | 2 | 3 | 93 | 5 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
112 | GREEN-024 | Largest Coset in 2A | Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha$. What is the largest size of coset guaranteed to be contained in $2A$? | This asks for the largest affine subspace (coset) contained in the doubling $2A = A + A$. Unlike the previous problem about many iterations, this focuses on just $2A$. Determining the optimal bound is a fundamental question in additive combinatorics over $\mathbb{F}_2^n$.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature... | 1 | open | null | null | 2 | 3 | 88 | 4 | 2024-01-01T00:00:00 | 2024-01-01T00:00:00 | true | {
"id": 2,
"name": "combinatorics",
"display_name": "Combinatorics",
"description": "Counting problems, graph theory, discrete structures.",
"slug": "combinatorics",
"order_index": 2,
"created_at": "2026-07-31T15:26:25.671Z"
} | {
"id": 1,
"level": 1,
"name": "L1: Tractable",
"description": "Problems that may be within reach with current techniques. Reserved for future additions.",
"color_class": "text-green-600 bg-green-50 border-green-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
113 | GREEN-025 | Additive Complements and Cosets | Suppose that $A \subset \mathbb{F}_2^n$ has an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$? | If $A$ has a small additive complement (a set $B$ with $A + B = \mathbb{F}_2^n$), does this force $2A$ to contain a large coset? This problem explores the relationship between additive complements and the structure of sumsets.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-17)
**Status:** part... | 2 | open | null | null | 2 | 3 | 91 | 5 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
114 | GREEN-026 | Partitions and Large Cosets | Suppose that $\mathbb{F}_2^n$ is partitioned into sets $A_1, \dots, A_K$. Does $2A_i$ contain a coset of codimension $O_K(1)$ for some $i$? | When partitioning a vector space into $K$ parts, at least one part must have substantial additive structure. This problem asks if one piece must have a doubling containing a large coset. It's a partitioning variant of the previous coset problems.
<!-- LITERATURE-TRIAGE:BEGIN -->
## Literature review (checked 2026-08-1... | 2 | open | null | null | 2 | 3 | 86 | 5 | 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": 2,
"level": 2,
"name": "L2: Intermediate",
"description": "Challenging problems requiring solid mathematical background. Reserved for future additions.",
"color_class": "text-blue-600 bg-blue-50 border-blue-200"
} | {
"id": 3,
"name": "green_problems",
"display_name": "Ben Green's 100 Open Problems",
"description": "A collection of 100 open problems in additive combinatorics and related areas, compiled by Ben Green.",
"slug": "green-problems",
"order_index": 3,
"created_at": "2026-07-31T15:26:25.671Z"
} | null | null | null | null | null | null | null | null | null | null | null | null | null |
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