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ALG-024 | The Bounded Burnside Problem | For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2, 5)$ finite? | The Burnside problem, posed in 1902, asks whether a finitely generated group in which every element has finite order must itself be finite. The bounded version restricts to groups where all elements have order dividing a fixed $n$. Major breakthroughs came when Novikov and Adian (1968) proved $B(m,n)$ is infinite for o... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 687 | 52 | train |
ALG-025 | The Guralnick-Thompson Conjecture | What are the composition factors of finite groups appearing in genus-0 systems? | This conjecture, proposed by Robert Guralnick and John Thompson, concerns the classification of finite groups that can act on Riemann surfaces of genus 0. The conjecture provides a list of simple groups that can appear as composition factors of such groups. The problem connects group theory with algebraic geometry and ... | open | L4: Expert | 4 | Algebra | null | null | null | 298 | 19 | train |
ALG-026 | The Herzog-Schönheim Conjecture | If a finite system of left cosets of subgroups of a group $G$ partitions $G$, must some two subgroups have the same index? | Proposed independently by Marcel Herzog and Jochanan Schönheim in 1974, this conjecture states that if finitely many left cosets of subgroups partition a group, then at least two of the subgroups must have the same finite index. This problem arises naturally in the study of group coverings and has connections to number... | open | L4: Expert | 4 | Algebra | null | null | null | 321 | 22 | train |
ALG-027 | The Inverse Galois Problem | Is every finite group the Galois group of some Galois extension of $\mathbb{Q}$? | The inverse Galois problem is one of the central open problems in Galois theory. While classical Galois theory establishes a correspondence between field extensions and groups, the inverse problem asks whether every finite group can be realized as the Galois group of an extension of the rational numbers. The problem wa... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 892 | 67 | train |
ALG-028 | The Isomorphism Problem for Coxeter Groups | Is there an algorithm to determine whether two Coxeter groups given by presentations are isomorphic? | Coxeter groups are fundamental objects in geometric group theory, generated by reflections with certain relations. They include the symmetry groups of regular polytopes and tessellations. The isomorphism problem asks whether there exists an algorithmic procedure to decide if two Coxeter groups, given by their Coxeter d... | open | L4: Expert | 4 | Algebra | null | null | null | 367 | 25 | train |
ALG-029 | Infinitude of Leinster Groups | Are there infinitely many Leinster groups? | A Leinster group is a finite group whose order equals the sum of the orders of its proper normal subgroups. Named after Tom Leinster who studied them in 1996, only two examples are currently known: the cyclic group of order 6 and a group of order 12. The question of whether infinitely many such groups exist remains ope... | open | L3: Advanced | 3 | Algebra | null | null | null | 245 | 18 | train |
ALG-031 | Finiteness of Finitely Presented Periodic Groups | Is every finitely presented periodic group finite? | A periodic group (or torsion group) is one in which every element has finite order. The question of whether a finitely presented periodic group must be finite was a major open problem for much of the 20th century. The restricted Burnside problem, solved by Zel'manov, showed that finitely generated groups where all elem... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 456 | 33 | train |
ALG-032 | The Surjunctivity Conjecture | Is every group surjunctive? | A group is surjunctive if every injective cellular automaton over that group is also surjective. Equivalently, every injective endomorphism of the shift space is surjective. This property was introduced by Gottschalk in 1973 and connects symbolic dynamics, cellular automata theory, and group theory. Gromov and Weiss pr... | open | L4: Expert | 4 | Algebra | null | null | null | 389 | 27 | train |
ALG-033 | The Sofic Groups Conjecture | Is every discrete countable group sofic? | A group is sofic if it can be approximated by finite symmetric groups in a precise sense. The concept was introduced by Gromov and Weiss around 1999 and has become central in modern group theory. All known groups are sofic: amenable groups, residually finite groups, linear groups, and many others. The soficity of all g... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 612 | 48 | train |
ALG-034 | Arthur's Conjectures | What is the structure of the discrete spectrum of automorphic forms on reductive groups? | Proposed by James Arthur in the 1980s, these conjectures describe the decomposition of the space of automorphic forms into irreducible representations. They provide a framework for understanding the discrete spectrum in terms of endoscopic groups and Arthur packets. The conjectures connect representation theory, harmon... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 478 | 35 | train |
ALG-035 | Dade's Conjecture | Is there a relationship between the numbers of irreducible characters in blocks of a finite group and its local subgroups? | Proposed by Everett Dade in 1992, this conjecture concerns the modular representation theory of finite groups. It relates the number of irreducible characters of a given defect in a block of a finite group to corresponding numbers in blocks of certain local subgroups (normalizers of p-subgroups). The conjecture is part... | open | L4: Expert | 4 | Algebra | null | null | null | 312 | 21 | train |
ALG-036 | The Demazure Conjecture | Can representations of semisimple algebraic groups be characterized over the integers? | Proposed by Michel Demazure in the 1970s, this conjecture concerns the existence of certain integral structures on representations of algebraic groups. It asks whether irreducible representations of semisimple algebraic groups over fields of positive characteristic can be deformed to characteristic zero while preservin... | open | L4: Expert | 4 | Algebra | null | null | null | 289 | 18 | train |
GEO-012 | The Spherical Bernstein Problem | What is the classification of complete minimal hypersurfaces in spheres of all dimensions? | This is a generalization of Bernstein's problem (solved by 1968) which asked whether the only minimal graph over all of Euclidean space is a hyperplane. The spherical version asks for the classification of complete minimal hypersurfaces in the sphere $S^{n+1}$. While progress has been made in specific dimensions, a com... | open | L4: Expert | 4 | Geometry | null | null | null | 387 | 24 | train |
GEO-013 | The Carathéodory Conjecture | Does every convex, closed, twice-differentiable surface in $\mathbb{R}^3$ have at least two umbilical points? | Proposed by Constantin Carathéodory in the 1920s, this conjecture concerns umbilical points on convex surfaces—points where the principal curvatures are equal. The conjecture states that any smooth closed convex surface in 3-dimensional Euclidean space must have at least two such points. A sphere has infinitely many um... | open | L4: Expert | 4 | Geometry | null | null | null | 456 | 31 | train |
GEO-014 | The Cartan-Hadamard Conjecture | Does the isoperimetric inequality hold for Cartan-Hadamard manifolds? | The classical isoperimetric inequality states that among all regions of fixed volume in Euclidean space, a ball has the smallest surface area. The Cartan-Hadamard conjecture asks whether this extends to Cartan-Hadamard manifolds—complete, simply connected Riemannian manifolds of nonpositive sectional curvature. The con... | open | L4: Expert | 4 | Geometry | null | null | null | 523 | 39 | train |
GEO-015 | Chern's Affine Conjecture | Does the Euler characteristic of a compact affine manifold vanish? | Proposed by Shiing-Shen Chern, this conjecture states that every closed affine manifold (a manifold with an atlas whose transition functions are affine transformations) has Euler characteristic zero. An affine structure is stronger than a smooth structure but weaker than a Riemannian structure. The conjecture has been ... | open | L4: Expert | 4 | Geometry | null | null | null | 398 | 27 | train |
GEO-016 | Chern's Conjecture for Hypersurfaces in Spheres | What minimal hypersurfaces in spheres have constant mean curvature? | This is actually a family of related conjectures proposed by Shiing-Shen Chern concerning the classification of minimal and constant mean curvature hypersurfaces embedded in spheres. One version asks whether the only minimal hypersurface in $S^{n+1}$ with constant scalar curvature is the totally geodesic $S^n$. These c... | open | L4: Expert | 4 | Geometry | null | null | null | 367 | 23 | train |
GEO-017 | The Closed Curve Problem | What are necessary and sufficient conditions for an integral curve defined by two periodic functions to be closed? | This problem asks for explicit, computable conditions to determine when a curve defined parametrically by integrating two periodic functions with the same period will close up. The question arises naturally in dynamical systems, Hamiltonian mechanics, and the study of periodic orbits. While special cases are understood... | open | L3: Advanced | 3 | Geometry | null | null | null | 289 | 19 | train |
GEO-018 | The Filling Area Conjecture | Does a hemisphere have minimum area among shortcut-free surfaces with a given boundary length? | This conjecture in systolic geometry states that among all surfaces in Euclidean space whose boundary is a closed curve of given length and which contain no shortcuts (the surface distance between boundary points equals the Euclidean distance), the hemisphere has minimal area. The problem was proposed by Gromov and con... | open | L4: Expert | 4 | Geometry | null | null | null | 334 | 22 | train |
GEO-019 | The Hopf Conjectures | What is the relationship between curvature and Euler characteristic for even-dimensional Riemannian manifolds? | Heinz Hopf proposed several conjectures relating the sign of sectional curvature to the Euler characteristic and other topological invariants of closed Riemannian manifolds. The most famous asks whether a closed even-dimensional manifold with positive (or negative) sectional curvature must have positive Euler character... | open | L5: Millennium Prize | 5 | Geometry | null | null | null | 567 | 43 | train |
GEO-021 | Yau's Conjecture on First Eigenvalues | Is the first eigenvalue of the Laplace-Beltrami operator on a minimal hypersurface in $S^{n+1}$ equal to $n$? | Proposed by Shing-Tung Yau, this conjecture states that for any closed embedded minimal hypersurface in the $(n+1)$-dimensional sphere $S^{n+1}$, the first nonzero eigenvalue of the Laplace-Beltrami operator equals $n$. This would provide a sharp spectral characterization of minimal hypersurfaces in spheres. The conjec... | open | L4: Expert | 4 | Geometry | null | null | null | 478 | 34 | train |
GEO-022 | The Hadwiger Covering Conjecture | Can every $n$-dimensional convex body be covered by at most $2^n$ smaller homothetic copies? | Proposed by Hugo Hadwiger in 1957, this conjecture states that any $n$-dimensional convex body can be covered by at most $2^n$ positive homothetic (scaled and translated) copies of itself with smaller ratio. The conjecture is known to be true for $n = 1$ (trivial) and $n = 2$ (proven), but remains open for $n \geq 3$. ... | open | L4: Expert | 4 | Geometry | null | null | null | 523 | 38 | train |
GEO-023 | The Happy Ending Problem | What is the minimum number of points in the plane needed to guarantee a convex $n$-gon? | The Happy Ending problem, named by Paul Erdős because it led to the marriage of Esther Klein and George Szekeres, asks for $g(n)$—the smallest number such that any set of $g(n)$ points in general position contains $n$ points forming a convex $n$-gon. It's known that $2^{n-2} + 1 \leq g(n) \leq \binom{2n-4}{n-2} + 1$. T... | open | L4: Expert | 4 | Geometry | null | null | null | 612 | 47 | train |
GEO-024 | The Heilbronn Triangle Problem | What is the largest minimum area of a triangle determined by $n$ points in a unit square? | Proposed by Hans Heilbronn in 1908, this problem asks how to place $n$ points in a unit square to maximize the smallest area of any triangle they determine. Heilbronn originally conjectured the maximum was $O(1/n^2)$, but this was disproven—the actual order is between $\Omega(\log n / n^2)$ and $O(1/n^{8/7-\epsilon})$.... | open | L4: Expert | 4 | Geometry | null | null | null | 445 | 31 | train |
GEO-026 | The Unit Distance Problem | What is the maximum number of unit distances determined by $n$ points in the plane? | This problem, posed by Erdős in 1946, asks for the maximum number of pairs of points at distance exactly 1 in a set of $n$ points in the Euclidean plane. The best known construction gives $\Omega(n^{4/3})$ unit distances, while the best upper bound is $O(n^{4/3})$. Determining the exact asymptotic (and whether the expo... | open | L4: Expert | 4 | Geometry | null | null | null | 567 | 42 | train |
GEO-028 | Ehrhart's Volume Conjecture | Does a convex body in $\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$? | Proposed by Eugène Ehrhart, this conjecture concerns lattice polytopes—convex bodies whose vertices have integer coordinates. It states that if a convex body in $n$ dimensions contains exactly one lattice point in its interior (which is its center of mass), then its volume cannot exceed $(n+1)^n/n!$, the volume of a re... | open | L4: Expert | 4 | Geometry | null | null | null | 389 | 27 | train |
ALG-039 | The Cherlin-Zilber Conjecture | Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field? | Proposed by Gregory Cherlin and Boris Zilber in the 1970s, this conjecture connects model theory and group theory. It states that any infinite simple group whose first-order theory is stable must be isomorphic to a simple algebraic group defined over an algebraically closed field. The conjecture has been verified for m... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 412 | 29 | train |
ALG-040 | The Generalized Star Height Problem | Can all regular languages be expressed with generalized regular expressions of bounded star height? | This problem in formal language theory asks whether there exists a uniform bound on the nesting depth of Kleene star operations needed to express any regular language using generalized regular expressions (which allow complementation). While the ordinary star height problem (without complementation) was solved—showing ... | open | L4: Expert | 4 | Algebra | null | null | null | 334 | 23 | train |
NT-031 | Hilbert's Tenth Problem for Number Fields | For which number fields is there an algorithm to determine solvability of Diophantine equations? | Hilbert's tenth problem asked for an algorithm to determine whether a Diophantine equation has integer solutions. Matiyasevich (building on work by Davis, Putnam, and Robinson) proved in 1970 that no such algorithm exists for the integers. The problem remains open for other rings, particularly number fields (finite ext... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 523 | 39 | train |
GEO-029 | Borsuk's Conjecture | Can every bounded set in $\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter? | Proposed by Karol Borsuk in 1933, this conjecture asks whether every bounded set in $n$-dimensional Euclidean space can be partitioned into $n+1$ parts, each with diameter strictly smaller than the original set. The conjecture held for dimensions up to 3 until 1993, when Kahn and Kalai found a counterexample in dimensi... | open | L4: Expert | 4 | Geometry | null | null | null | 523 | 39 | train |
GEO-030 | The Kissing Number Problem | What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions? | The kissing number $\tau_n$ is the maximum number of non-overlapping unit spheres that can simultaneously touch a central unit sphere in $n$-dimensional Euclidean space. Known exactly only for dimensions 1, 2, 3, 4, 8, and 24, this problem has connections to sphere packing, coding theory, and lattice theory. The dimens... | open | L4: Expert | 4 | Geometry | null | null | null | 612 | 46 | train |
GEO-031 | Ulam's Packing Conjecture | Is the sphere the worst-packing convex solid? | Proposed by Stanisław Ulam, this conjecture asks which three-dimensional convex body has the smallest packing density. Ulam conjectured that the sphere is the worst-packing convex solid, meaning that among all convex bodies in 3D, spheres have the smallest proportion of space filled when packed. While the sphere packin... | open | L4: Expert | 4 | Geometry | null | null | null | 445 | 32 | train |
GEO-032 | Sphere Packing in High Dimensions | What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24? | The sphere packing problem asks for the densest arrangement of non-overlapping unit spheres in $n$-dimensional Euclidean space. Solved for dimensions 1 and 2 (trivial), dimension 3 by Hales (1998, computer-assisted proof), dimension 8 by Viazovska (2016), and dimension 24 by Cohn et al. (2016), the problem remains open... | open | L5: Millennium Prize | 5 | Geometry | null | null | null | 734 | 58 | train |
COMB-010 | The Cap Set Problem | What is the maximum size of a cap set in $\mathbb{F}_3^n$? | A cap set is a subset of the $n$-dimensional vector space over the three-element field with no three elements in arithmetic progression (analogous to the card game SET). The problem asks for the maximum size of such a set as a function of $n$. In 2016, Ellenberg and Gijswijt proved an upper bound of $O(2.756^n)$, drama... | open | L4: Expert | 4 | Combinatorics | null | null | null | 523 | 40 | train |
COMB-013 | Ramsey Number $R(5,5)$ | What is the exact value of the Ramsey number $R(5,5)$? | Ramsey numbers quantify the size at which complete disorder becomes impossible. $R(5,5)$ is the minimum number of vertices such that any two-coloring of the edges of the complete graph contains either a red $K_5$ or a blue $K_5$. It is known that $43 \leq R(5,5) \leq 48$, but the exact value remains unknown despite ove... | open | L4: Expert | 4 | Combinatorics | null | null | null | 823 | 67 | train |
NT-032 | Gauss Circle Problem | How far can the number of lattice points in a circle centered at the origin deviate from the area of the circle? | The Gauss circle problem asks for the tightest bound on the error term in counting integer lattice points $(m,n)$ inside a circle of radius $r$ centered at the origin. The number of such points is $\pi r^2 + E(r)$ where $E(r)$ is the error. It is known that $E(r) = O(r^{2/3})$ and $E(r) = \Omega(r^{1/2} \log r)$, but t... | open | L4: Expert | 4 | Number Theory | null | null | null | 478 | 35 | train |
NT-033 | Grimm's Conjecture | Can each element of a set of consecutive composite numbers be assigned a distinct prime divisor? | Proposed by C. A. Grimm in 1969, this conjecture states that if we have $k$ consecutive composite numbers, then there exist $k$ distinct primes each dividing one of these numbers. For example, the consecutive composites $24, 25, 26, 27, 28$ have distinct prime divisors $3, 5, 13, 7, 2$ respectively. While verified comp... | open | L4: Expert | 4 | Number Theory | null | null | null | 412 | 29 | train |
NT-034 | Hall's Conjecture | For any $\varepsilon > 0$, is there a constant $c(\varepsilon)$ such that either $y^2 = x^3$ or $|y^2 - x^3| > c(\varepsilon) x^{1/2-\varepsilon}$? | Proposed by Marshall Hall Jr. in 1970, this conjecture provides a measure of how close a perfect square can be to a perfect cube without being equal. It strengthens earlier work on Diophantine approximation and relates to the ABC conjecture. The conjecture has been verified for many special cases but remains open in ge... | open | L4: Expert | 4 | Number Theory | null | null | null | 445 | 33 | train |
NT-035 | Lehmer's Totient Problem | If Euler's totient function $\phi(n)$ divides $n-1$, must $n$ be prime? | Posed by D. H. Lehmer in 1932, this problem asks whether any composite number $n$ exists such that $\phi(n)$ divides $n-1$, where $\phi(n)$ counts integers up to $n$ coprime to $n$. For all primes $p$, we have $\phi(p) = p-1$, so the divisibility holds. Lehmer conjectured no composite number has this property. It has b... | open | L4: Expert | 4 | Number Theory | null | null | null | 523 | 41 | train |
NT-036 | Magic Square of Squares | Does there exist a 3×3 magic square composed entirely of distinct perfect squares? | A magic square has the property that all rows, columns, and diagonals sum to the same value. While magic squares of integers are well understood, the question of whether a 3×3 magic square can be constructed using only distinct perfect squares has remained open for centuries. Martin LaBar proved in 1984 that no such sq... | open | L4: Expert | 4 | Number Theory | null | null | null | 589 | 47 | train |
NT-037 | Mahler's 3/2 Problem | Is there a real number $x$ such that the fractional parts of $x(3/2)^n$ are all less than $1/2$ for every positive integer $n$? | Proposed by Kurt Mahler in the 1960s, this problem concerns the distribution of the sequence $\{x(3/2)^n\}$ modulo 1, where $\{y\}$ denotes the fractional part of $y$. Mahler conjectured that no such $x$ exists. The problem relates to ergodic theory, uniform distribution, and Diophantine approximation. While various pa... | open | L4: Expert | 4 | Number Theory | null | null | null | 398 | 28 | train |
NT-038 | Newman's Conjecture | Does the partition function satisfy any arbitrary congruence infinitely often? | Proposed by Morris Newman, this conjecture concerns the partition function $p(n)$, which counts the number of ways to write $n$ as a sum of positive integers. Newman conjectured that for any integers $a$ and $m$ with $\gcd(a,m) = 1$, there are infinitely many $n$ such that $p(n) \equiv a \pmod{m}$. This would imply the... | open | L4: Expert | 4 | Number Theory | null | null | null | 367 | 26 | train |
NT-039 | Scholz Conjecture | Is the shortest addition chain for $2^n - 1$ at most $n - 1$ plus the length of the shortest addition chain for $n$? | An addition chain for $m$ is a sequence $1 = a_0 < a_1 < \cdots < a_r = m$ where each $a_i$ (for $i > 0$) is the sum of two earlier terms. Scholz conjectured in 1937 that $\ell(2^n-1) \leq n-1+\ell(n)$ where $\ell(m)$ denotes the minimum length of an addition chain for $m$. This has applications to efficient exponentia... | open | L4: Expert | 4 | Number Theory | null | null | null | 412 | 30 | train |
NT-041 | Infinitely Many Perfect Numbers | Are there infinitely many perfect numbers? | A perfect number equals the sum of its proper divisors (divisors excluding itself). Examples include 6 = 1+2+3 and 28 = 1+2+4+7+14. Euclid proved that if $2^p - 1$ is prime (a Mersenne prime), then $2^{p-1}(2^p-1)$ is perfect. All known perfect numbers have this form and are even. Whether infinitely many exist depends ... | open | L4: Expert | 4 | Number Theory | null | null | null | 678 | 54 | train |
NT-043 | Quasiperfect Numbers | Do quasiperfect numbers exist? | A quasiperfect number is a natural number $n$ such that the sum of its divisors equals $2n + 1$ (one more than twice the number). No quasiperfect number has ever been found. It has been proven that if one exists, it must be an odd square number greater than $10^{35}$, and have at least seven distinct prime factors. The... | open | L4: Expert | 4 | Number Theory | null | null | null | 398 | 28 | train |
NT-044 | Almost Perfect Numbers Beyond Powers of 2 | Do any almost perfect numbers exist that are not powers of 2? | An almost perfect number $n$ has the sum of its proper divisors equal to $n - 1$. All powers of 2 are almost perfect, since the divisors of $2^k$ are $1, 2, 4, \ldots, 2^{k-1}$ which sum to $2^k - 1$. It remains unknown whether any odd almost perfect number exists, or any even almost perfect number that is not a power ... | open | L4: Expert | 4 | Number Theory | null | null | null | 356 | 25 | train |
NT-045 | The Number of Idoneal Numbers | Are there exactly 65 idoneal numbers, or could there be 66 or 67? | Idoneal numbers (also called suitable or convenient numbers) are positive integers $D$ such that if $n = ax^2 + by^2$ with coprime $a,b$ is uniquely representable, then $n$ is a prime power or twice a prime power. Euler conjectured 65 such numbers exist, the largest being 1848. Weinberger proved in 1973 that at most on... | open | L4: Expert | 4 | Number Theory | null | null | null | 334 | 24 | train |
NT-048 | Infinitely Many Giuga Numbers | Are there infinitely many Giuga numbers? | A Giuga number is a composite number $n$ such that $p$ divides $(n/p - 1)$ for every prime divisor $p$ of $n$. Equivalently, $\sum_{p|n} (1/p) - 1/n$ is an integer. Only 15 Giuga numbers are known, the smallest being 30. Giuga conjectured that if $1 + \sum_{i=1}^{n-1} i^{n-1} \equiv 0 \pmod{n}$ for composite $n$, then ... | open | L4: Expert | 4 | Number Theory | null | null | null | 367 | 26 | train |
NT-049 | Lychrel Numbers in Base 10 | Do Lychrel numbers exist in base 10? | A Lychrel number is a natural number that never forms a palindrome through the iterative process of adding it to its reverse. For example, 89 is not Lychrel: 89 + 98 = 187, 187 + 781 = 968, 968 + 869 = 1837, 1837 + 7381 = 9218, 9218 + 8129 = 17347, 17347 + 74371 = 91718, 91718 + 81719 = 173437, 173437 + 734371 = 907808... | open | L3: Advanced | 3 | Number Theory | null | null | null | 512 | 39 | train |
NT-050 | Odd Weird Numbers | Do any odd weird numbers exist? | A weird number is a natural number that is abundant (the sum of its proper divisors exceeds the number) but not semiperfect (no subset of its divisors sums to the number). The smallest weird number is 70. All known weird numbers are even, and it has been conjectured that no odd weird numbers exist. If an odd weird numb... | open | L4: Expert | 4 | Number Theory | null | null | null | 378 | 27 | train |
NT-051 | Normality of Pi | Is $\pi$ a normal number in base 10? | A number is normal in base 10 if every digit 0-9 appears with equal frequency (1/10) in its decimal expansion, and more generally, every sequence of $k$ digits appears with frequency $1/10^k$. While the digits of $\pi$ appear statistically random in computational tests extending to trillions of digits, no proof of norm... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 823 | 68 | train |
NT-052 | Normality of Irrational Algebraic Numbers | Are all irrational algebraic numbers normal in every base? | An algebraic number is a root of a polynomial with integer coefficients. Normal numbers have every digit sequence appear with the expected frequency in their base expansions. It is conjectured that all irrational algebraic numbers like $\sqrt{2}$ are normal in every integer base, but not a single irrational algebraic n... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 567 | 45 | train |
NT-053 | Is 10 a Solitary Number? | Is 10 a solitary number (no other number shares its abundancy index)? | The abundancy index of $n$ is $\sigma(n)/n$ where $\sigma(n)$ is the sum of divisors of $n$. A number is solitary if no other number has the same abundancy index. For 10, we have $\sigma(10) = 1+2+5+10 = 18$, giving abundancy $18/10 = 9/5$. It remains unknown whether any other number has abundancy $9/5$. Numbers in ami... | open | L3: Advanced | 3 | Number Theory | null | null | null | 334 | 24 | train |
NT-055 | Erdős Conjecture on Arithmetic Progressions | If the sum of reciprocals of a set of positive integers diverges, does the set contain arbitrarily long arithmetic progressions? | Erdős conjectured that if $A \subseteq \mathbb{N}$ and $\sum_{a \in A} 1/a = \infty$, then $A$ contains arithmetic progressions of arbitrary length. This strengthens Szemerédi's theorem, which only requires positive density. The conjecture remains open even for progressions of length 3. In 2020, Bloom and Sisask made m... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 534 | 42 | train |
NT-056 | Erdős-Turán Conjecture on Additive Bases | If $B$ is an additive basis of order 2, must the representation function tend to infinity? | An additive basis of order 2 is a set $B$ such that every sufficiently large integer can be written as the sum of two elements of $B$. The representation function $r_B(n)$ counts the number of ways to write $n$ as $b_1 + b_2$ with $b_1, b_2 \in B$. Erdős and Turán conjectured in 1941 that if $B$ is an additive basis of... | open | L4: Expert | 4 | Number Theory | null | null | null | 456 | 34 | train |
NT-058 | Lander-Parkin-Selfridge Conjecture | If the sum of $m$ $k$-th powers equals the sum of $n$ $k$-th powers, must $m + n \geq k$? | This conjecture generalizes Fermat's Last Theorem to sums of powers. It states that if $a_1^k + \cdots + a_m^k = b_1^k + \cdots + b_n^k$ with positive integers and the two sums are different, then $m + n \geq k$. Euler conjectured the stronger statement that at least $k$ $k$-th powers are needed, but this was disproved... | open | L4: Expert | 4 | Number Theory | null | null | null | 489 | 37 | train |
NT-059 | Lemoine's Conjecture | Can every odd integer greater than 5 be expressed as the sum of an odd prime and an even semiprime? | Proposed by Émile Lemoine in 1894, this conjecture states that every odd number $n > 5$ can be written as $n = p + 2q$ where $p$ and $q$ are primes. An even semiprime is twice a prime. For example, $27 = 13 + 2(7)$, $31 = 19 + 2(6)$ is invalid since 6 isn't prime, but $31 = 5 + 2(13)$ works. This is weaker than Goldbac... | open | L4: Expert | 4 | Number Theory | null | null | null | 445 | 33 | train |
NT-060 | Recamán's Sequence Completeness | Does every nonnegative integer appear in Recamán's sequence? | Recamán's sequence starts with $a_0 = 0$ and follows the rule: $a_n = a_{n-1} - n$ if that value is positive and not already in the sequence, otherwise $a_n = a_{n-1} + n$. This produces: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, ... Named after Colombian mathematician Bernardo Recamán Santos, this sequence has... | open | L3: Advanced | 3 | Number Theory | null | null | null | 512 | 40 | train |
NT-061 | Skolem Problem | Can an algorithm determine if a constant-recursive sequence contains a zero? | A constant-recursive sequence satisfies a linear recurrence with constant coefficients, like the Fibonacci sequence. The Skolem problem asks whether there exists an algorithm to determine if such a sequence ever equals zero. This is known to be decidable for sequences of order up to 4, but the general problem remains o... | open | L4: Expert | 4 | Number Theory | null | null | null | 389 | 28 | train |
NT-062 | Waring's Problem: Exact Values | What are the exact values of $g(k)$ and $G(k)$ for all $k$ in Waring's problem? | Waring's problem concerns representing integers as sums of $k$-th powers. Let $g(k)$ be the minimum number such that every positive integer can be written as a sum of at most $g(k)$ $k$-th powers, allowing any number of terms. Let $G(k)$ be the same but excluding a finite set of exceptions. We know $g(2)=4$ (Lagrange),... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 567 | 44 | train |
NT-063 | Density of Ulam Numbers | Do the Ulam numbers have a positive density? | The Ulam numbers start with 1, 2, and each subsequent number is the smallest integer that can be expressed as the sum of two distinct earlier Ulam numbers in exactly one way: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ... Named after Stanisław Ulam, these numbers appear to have density around 0.07, but whether the density exist... | open | L4: Expert | 4 | Number Theory | null | null | null | 398 | 29 | train |
NT-064 | Class Number Problem | Are there infinitely many real quadratic number fields with unique factorization? | A number field has unique factorization if every nonzero element factors uniquely into irreducibles. For real quadratic fields $\mathbb{Q}(\sqrt{d})$ with $d > 0$ square-free, unique factorization is equivalent to having class number 1. Gauss conjectured infinitely many such fields exist. While infinitely many imaginar... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 478 | 36 | train |
NT-065 | Hilbert's Twelfth Problem | Can the Kronecker-Weber theorem on abelian extensions of $\mathbb{Q}$ be extended to any base number field? | The Kronecker-Weber theorem states that every abelian extension of the rationals $\mathbb{Q}$ is contained in a cyclotomic field (generated by roots of unity). Hilbert's 12th problem asks for an analogous explicit construction of abelian extensions of arbitrary number fields. For imaginary quadratic fields, complex mul... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 512 | 40 | train |
NT-066 | Leopoldt's Conjecture | Does the $p$-adic regulator of an algebraic number field not vanish? | Leopoldt's conjecture, proposed in 1962, states that the $p$-adic regulator of an algebraic number field $K$ is nonzero for every prime $p$. The regulator measures the "size" of the unit group. The conjecture has been verified for abelian extensions of $\mathbb{Q}$ and many other special cases, but remains open in gene... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 389 | 29 | train |
NT-067 | Lindelöf Hypothesis | For all $\varepsilon > 0$, does $\zeta(1/2 + it) = o(t^\varepsilon)$ as $t \to \infty$? | The Lindelöf hypothesis concerns the growth rate of the Riemann zeta function $\zeta(s)$ on the critical line $\text{Re}(s) = 1/2$. It states that for any $\varepsilon > 0$, we have $|\zeta(1/2 + it)| = o(t^\varepsilon)$. This is weaker than the Riemann Hypothesis but still unproven. The best known bound is $O(t^{13/84... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 545 | 43 | train |
NT-069 | Grand Riemann Hypothesis | Do all automorphic L-functions have their nontrivial zeros on the critical line? | The Grand Riemann Hypothesis extends RH to all automorphic L-functions, a vast class including Dirichlet L-functions, Dedekind zeta functions, and L-functions of modular forms. It asserts that all nontrivial zeros lie on the critical line $\text{Re}(s) = 1/2$. This would have profound consequences for prime distributio... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 712 | 59 | train |
NT-070 | Montgomery's Pair Correlation Conjecture | Does the pair correlation function of Riemann zeta zeros match that of random Hermitian matrices? | Montgomery conjectured in 1973 that the statistical distribution of gaps between zeros of the Riemann zeta function matches the pair correlation of eigenvalues from the Gaussian Unitary Ensemble (GUE) of random matrix theory. This remarkable connection between number theory and quantum physics was discovered through nu... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 567 | 46 | train |
NT-071 | Dirichlet's Divisor Problem | What is the optimal exponent in the error term for the divisor summatory function? | Let $D(x) = \sum_{n \leq x} d(n)$ where $d(n)$ counts the divisors of $n$. Dirichlet proved $D(x) = x \log x + (2\gamma - 1)x + \Delta(x)$ where $\gamma$ is Euler's constant and $\Delta(x)$ is the error. The problem asks for the infimum $\theta$ such that $\Delta(x) = O(x^\theta)$. It is known that $1/4 \leq \theta < 1... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 445 | 34 | train |
GEO-033 | Erdős-Ulam Problem | Is there a dense set of points in the plane with all pairwise distances rational? | Proposed by Paul Erdős and Stanisław Ulam, this problem asks whether there exists a dense subset of the Euclidean plane (dense in the usual topology) such that the distance between any two points is a rational number. While finite and countable dense sets with rational distances are known (like rational points on a cir... | open | L4: Expert | 4 | Geometry | null | null | null | 478 | 36 | train |
NT-073 | Four Exponentials Conjecture | If $x_1, x_2$ are linearly independent over $\mathbb{Q}$ and $y_1, y_2$ are linearly independent over $\mathbb{Q}$, is at least one of $e^{x_1 y_1}, e^{x_1 y_2}, e^{x_2 y_1}, e^{x_2 y_2}$ transcendental? | This conjecture, a consequence of Schanuel's conjecture, asserts that under the stated conditions, at least one of the four exponentials must be transcendental. The six exponentials theorem (proven) states that if $x_1, x_2, x_3$ are $\mathbb{Q}$-linearly independent and $y_1, y_2$ are $\mathbb{Q}$-linearly independent... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 445 | 34 | train |
NT-074 | Irrationality of Euler's Constant | Is the Euler-Mascheroni constant $\gamma$ irrational? | Euler's constant $\gamma = \lim_{n \to \infty} (1 + 1/2 + 1/3 + \cdots + 1/n - \ln n) \approx 0.5772$ appears throughout mathematics but its arithmetic nature remains mysterious. It is not even known whether $\gamma$ is irrational, let alone transcendental. While computational evidence suggests irrationality (verified ... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 712 | 58 | train |
NT-075 | Transcendence of Apéry's Constant | Is $\zeta(3) = 1 + 1/8 + 1/27 + 1/64 + \cdots$ transcendental? | Apéry's constant $\zeta(3) \approx 1.202$ is the value of the Riemann zeta function at 3. Roger Apéry proved its irrationality in 1978 using ingenious continued fraction methods, surprising the mathematical community. Whether $\zeta(3)$ is transcendental remains unknown. More generally, the transcendence of $\zeta(2k+1... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 589 | 47 | train |
NT-076 | Littlewood Conjecture | For any two real numbers $\alpha, \beta$, does $\liminf_{n \to \infty} n \|n\alpha\| \|n\beta\| = 0$? | Proposed by John Edensor Littlewood around 1930, where $\|x\|$ denotes the distance from $x$ to the nearest integer. The conjecture asserts a simultaneous approximation property: for any pair of real numbers, infinitely many integers $n$ exist such that both $n\alpha$ and $n\beta$ are simultaneously close to integers, ... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 456 | 35 | train |
NT-077 | Integer Factorization in Polynomial Time | Can integer factorization be solved in polynomial time on a classical computer? | The integer factorization problem asks: given a composite number $n$, find its prime factors. The best known classical algorithm (general number field sieve) runs in sub-exponential time $\exp(O((\ln n)^{1/3}(\ln \ln n)^{2/3}))$. Whether a polynomial-time classical algorithm exists is unknown and has profound implicati... | open | L4: Expert | 4 | Number Theory | null | null | null | 734 | 61 | train |
NT-078 | Beal's Conjecture | For $A^x + B^y = C^z$ with $x, y, z > 2$, must $A$, $B$, and $C$ share a common prime factor? | Proposed by banker and amateur mathematician Andrew Beal in 1993, this conjecture generalizes Fermat's Last Theorem. It asserts that if $A^x + B^y = C^z$ where $A, B, C, x, y, z$ are positive integers with $x, y, z > 2$, then $A$, $B$, and $C$ must have a common prime factor. For example, $3^3 + 6^3 = 3^5$ satisfies th... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 712 | 59 | train |
NT-084 | Bunyakovsky Conjecture | Does an irreducible integer polynomial with no fixed prime divisor produce infinitely many primes? | Proposed by Viktor Bunyakovsky in 1857, this generalizes Dirichlet's theorem on primes in arithmetic progressions. It states that if polynomial $f(x)$ has integer coefficients, positive leading coefficient, is irreducible over integers, and has no common prime divisor of all its values $f(n)$ for positive integers $n$,... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 512 | 41 | train |
NT-085 | Dickson's Conjecture | Do finitely many linear forms simultaneously take prime values infinitely often, barring congruence obstructions? | Proposed by Leonard Eugene Dickson in 1904, this generalizes Dirichlet's theorem and implies many prime conjectures. For linear forms $a_1 + b_1 n, \ldots, a_k + b_k n$ with each $b_i \geq 1$, if no congruence condition forces a composite, then infinitely many $n$ exist making all forms simultaneously prime. This would... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 445 | 34 | train |
NT-086 | Brocard's Conjecture (Prime Gaps) | Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$? | Proposed by Henri Brocard in 1904, this conjecture concerns the density of primes near perfect squares. For consecutive primes $p_n$ and $p_{n+1}$, Brocard conjectured there are always at least 4 primes in the interval $(p_n^2, p_{n+1}^2)$, except for the cases $(2^2, 3^2)$ which contains only one prime (5). Verified c... | open | L4: Expert | 4 | Number Theory | null | null | null | 398 | 29 | train |
NT-087 | Agoh-Giuga Conjecture | Is $p$ prime if and only if $pB_{p-1} \equiv -1 \pmod{p}$ for the Bernoulli number $B_{p-1}$? | This conjecture combines work of Takashi Agoh (1990) and Giuseppe Giuga (1950), providing a primality criterion via Bernoulli numbers. Bernoulli numbers $B_n$ appear in number theory and analysis. The conjecture states: $p$ is prime iff $pB_{p-1} \equiv -1 \pmod{p}$. The forward direction is known (if $p$ prime, the co... | open | L4: Expert | 4 | Number Theory | null | null | null | 334 | 25 | train |
NT-088 | Elliott-Halberstam Conjecture | Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)? | Proposed in 1968, this strengthens the Bombieri-Vinogradov theorem about primes in arithmetic progressions. For most moduli $q < x^\theta$, the primes are equidistributed among valid residue classes. Bombieri-Vinogradov proves this for $\theta < 1/2$. Elliott-Halberstam conjectures it holds for any $\theta < 1$. This w... | open | L5: Millennium Prize | 5 | Number Theory | null | null | null | 412 | 32 | train |
ALG-001 | Birch–Tate Conjecture | Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function? | The Birch–Tate conjecture connects algebraic K-theory to number theory. For a number field $F$, it relates the order of the center of the Steinberg group $\text{St}(\mathcal{O}_F)$ (where $\mathcal{O}_F$ is the ring of integers) to special values of the Dedekind zeta function $\zeta_F(s)$ at $s = -1$. The conjecture pr... | open | L5: Millennium Prize | 5 | Algebra | null | null | null | 245 | 18 | train |
ALG-004 | Crouzeix's Conjecture | Is $\|f(A)\| \leq 2\sup_{z \in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range? | Michel Crouzeix conjectured in 2004 that for any $n \times n$ complex matrix $A$ and any function $f$ analytic on the numerical range $W(A) = \{\langle Ax, x \rangle : \|x\| = 1\}$, the matrix norm satisfies $\|f(A)\| \leq 2\|f\|_{W(A)}$. The constant 2 is conjectured to be optimal. Crouzeix proved the bound with const... | open | L4: Expert | 4 | Algebra | null | null | null | 278 | 21 | train |
ALG-006 | Perfect Cuboid | Does there exist a rectangular cuboid with integer edges, face diagonals, and space diagonal? | A perfect cuboid would have integer values for all of: three edge lengths $a, b, c$, three face diagonals $\sqrt{a^2+b^2}, \sqrt{b^2+c^2}, \sqrt{c^2+a^2}$, and the space diagonal $\sqrt{a^2+b^2+c^2}$. This is the 3D generalization of the Pythagorean triple problem (which has infinitely many solutions). Despite extensiv... | open | L3: Advanced | 3 | Algebra | null | null | null | 423 | 35 | train |
ALG-009 | Zauner's Conjecture (SIC-POVM) | Do symmetric informationally complete POVMs exist in all dimensions? | Zauner's conjecture, central to quantum information theory, asks whether SIC-POVMs (Symmetric Informationally Complete Positive Operator-Valued Measures) exist in all finite-dimensional Hilbert spaces. A SIC-POVM in dimension $d$ consists of $d^2$ pure quantum states with pairwise fidelity $1/(d+1)$, forming a regular ... | open | L4: Expert | 4 | Algebra | null | null | null | 298 | 26 | train |
ALG-012 | Andrews–Curtis Conjecture | Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves? | The Andrews–Curtis conjecture, proposed in 1965, concerns group presentations. A balanced presentation has the same number of generators and relators. The trivial presentation is $\langle x \mid x \rangle$. Nielsen transformations on relators include: replacing relator $r$ with $r^{-1}$, with $rs$ for another relator $... | open | L4: Expert | 4 | Algebra | null | null | null | 289 | 23 | train |
ALG-014 | Herzog–Schönheim Conjecture | Can a finite system of left cosets forming a partition of a group have distinct indices? | The Herzog–Schönheim conjecture states: if left cosets $g_iH_i$ of subgroups $H_i$ partition a group $G$, then at least two indices $[G:H_i]$ must be equal. Equivalently, you cannot partition a group using cosets of subgroups with all different indices. The conjecture is verified for many cases: finite abelian groups, ... | open | L4: Expert | 4 | Algebra | null | null | null | 198 | 16 | train |
ANA-006 | Navier-Stokes Regularity | Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time? | One of the seven Millennium Prize Problems ($1M prize). The 3D Navier-Stokes equations govern fluid flow: $\partial_t u + (u \cdot \nabla)u = \nu \Delta u - \nabla p + f$ with $\nabla \cdot u = 0$. Given smooth initial conditions and forcing, do solutions remain smooth globally, or can finite-time singularities develop... | open | L5: Millennium Prize | 5 | Partial Differential Equations | null | null | null | 892 | 67 | train |
COMB-001 | 1/3–2/3 Conjecture | Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions? | For a finite partially ordered set (poset) that is not totally ordered, the 1/3–2/3 conjecture asks: do there always exist elements $x$ and $y$ such that the probability $x$ appears before $y$ in a uniformly random linear extension is strictly between 1/3 and 2/3? Linear extensions are total orderings consistent with t... | open | L4: Expert | 4 | Combinatorics | null | null | null | 234 | 19 | train |
COMB-002 | Lonely Runner Conjecture | If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\geq 1/k$ from others) at some time? | Proposed by J. M. Wills in 1967, this conjecture concerns runners on a unit-length circular track with distinct constant speeds. A runner is "lonely" if all other runners are at distance at least $1/k$ away. The conjecture states every runner is lonely at some time. Verified for $k \leq 7$ runners. The problem has refo... | open | L4: Expert | 4 | Combinatorics | null | null | null | 312 | 26 | train |
COMB-003 | Union-Closed Sets Conjecture | For a finite family of sets closed under unions, must some element appear in at least half the sets? | Frankl's union-closed sets conjecture (1979) states: if a finite family $\mathcal{F}$ of sets is closed under pairwise unions (i.e., $A, B \in \mathcal{F} \Rightarrow A \cup B \in \mathcal{F}$), then there exists an element appearing in at least $|\mathcal{F}|/2$ sets. The conjecture is verified for many special cases:... | open | L4: Expert | 4 | Combinatorics | null | null | null | 387 | 31 | train |
COMB-006 | Sunflower Conjecture | For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$? | Erdős and Rado (1960) defined an $r$-sunflower as a collection of $r$ sets $A_1, \ldots, A_r$ with common intersection $C$ (the core) such that the sets $A_i \setminus C$ are pairwise disjoint (the petals). Their theorem: any family of size-$k$ sets with at least $k! \cdot r^k$ members contains an $r$-sunflower. The su... | open | L4: Expert | 4 | Combinatorics | null | null | null | 367 | 29 | train |
GRAPH-003 | Cycle Double Cover Conjecture | Does every bridgeless graph have a collection of cycles covering each edge exactly twice? | The cycle double cover conjecture states: every bridgeless graph (no bridge edges) has a cycle double cover—a collection of cycles such that each edge appears in exactly two cycles. Proposed by Szekeres (1973) and Seymour (1979), this is equivalent to several other conjectures in graph theory. Known for planar graphs (... | open | L4: Expert | 4 | Graph Theory | null | null | null | 312 | 25 | train |
GRAPH-004 | Erdős–Hajnal Conjecture | For any fixed graph $H$, do $H$-free graphs contain large cliques or independent sets? | The Erdős–Hajnal conjecture (1977) asks: for any graph $H$, is there $\delta > 0$ such that every $n$-vertex graph with no induced copy of $H$ contains a clique or independent set of size at least $n^\delta$? This would dramatically strengthen Ramsey theory for hereditary graph classes. For general graphs, Ramsey theor... | open | L5: Millennium Prize | 5 | Graph Theory | null | null | null | 289 | 23 | train |
GRAPH-005 | Lovász Conjecture | Does every finite connected vertex-transitive graph have a Hamiltonian path? | Proposed by László Lovász in 1969, this conjecture states that every finite connected vertex-transitive graph (graph with transitive automorphism group) contains a Hamiltonian path. A stronger version asks for a Hamiltonian cycle. The conjecture is verified for Cayley graphs (Rapaport-Strasser, 1985 for primes; Marušič... | open | L4: Expert | 4 | Graph Theory | null | null | null | 267 | 21 | train |
GRAPH-006 | Hadwiger–Nelson Problem | What is the chromatic number of the plane with unit distance graph coloring? | The Hadwiger–Nelson problem asks: what is the minimum number of colors needed to color the plane such that no two points at distance exactly 1 have the same color? This is equivalent to finding the chromatic number of the unit distance graph in $\mathbb{R}^2$. It has been known since 1950 that $4 \leq \chi \leq 7$. In ... | open | L4: Expert | 4 | Graph Theory | null | null | null | 421 | 35 | train |
TOP-002 | Borel Conjecture | Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups? | The Borel conjecture states: if two aspherical closed manifolds (manifolds with contractible universal cover) have isomorphic fundamental groups, then they are homeomorphic. An aspherical manifold has all higher homotopy groups trivial, so its topology is determined by $\pi_1$. The conjecture is a topological rigidity ... | open | L5: Millennium Prize | 5 | Topology | null | null | null | 278 | 22 | train |
TOP-003 | Volume Conjecture | Do quantum invariants of knots relate asymptotically to hyperbolic volume? | The volume conjecture, proposed by Kashaev (1997) and generalized by Murakami-Murakami, relates quantum topology to hyperbolic geometry. For a hyperbolic knot $K$ in $S^3$, let $J_N(K; q)$ be the colored Jones polynomial at $q = e^{2\pi i/N}$. The conjecture states: $\lim_{N \to \infty} \frac{2\pi \log|J_N(K; e^{2\pi i... | open | L5: Millennium Prize | 5 | Topology | null | null | null | 245 | 19 | train |
TOP-004 | Novikov Conjecture | Are certain combinations of Pontryagin classes homotopy invariant? | The Novikov conjecture, proposed by Sergei Novikov in 1965, is a fundamental problem in topology and differential geometry. For a closed oriented manifold $M$ with fundamental group $\pi$, certain rational linear combinations of Pontryagin classes evaluated on the fundamental class should be homotopy invariants when pu... | open | L5: Millennium Prize | 5 | Topology | null | null | null | 312 | 25 | train |
GEOM-007 | Kakeya Conjecture | Must a Kakeya set in $\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$? | A Kakeya set in $\mathbb{R}^n$ is a compact set containing a unit line segment in every direction. The Kakeya conjecture states such sets must have full Hausdorff and Minkowski dimension $n$. In $\mathbb{R}^2$, Kakeya sets can have measure zero (Davies 1971) but must have Hausdorff dimension 2 (proven). For $n \geq 3$,... | open | L5: Millennium Prize | 5 | Geometry | null | null | null | 289 | 23 | train |
GEOM-008 | Illumination Problem | Can every convex body in $\mathbb{R}^n$ be illuminated by $2^n$ light sources? | The illumination problem (or Hadwiger's problem) asks: what is the minimum number of light sources (point sources or directions) needed to illuminate the entire boundary of any convex body in $\mathbb{R}^n$? A point on the boundary is illuminated if the ray from the light source to that point does not intersect the int... | open | L4: Expert | 4 | Geometry | null | null | null | 234 | 19 | train |
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