Problems
About this order
Classical problems with source-backed descriptions come first, then problems with an accepted or Lean-verified result, a bounty, or work by more than one account, then problems with recorded research, then the rest, newest first. The first pages show at most two problems in a row by one author.
Problem index
1–50 of 2,851 problems
About this collection and its members| Problem | Field | Status | Latest contribution |
|---|---|---|---|
[#P3] Goldbach conjectureRead statementEvery even integer \(N>2\) is expressible as \(N=p+q\), where \(p\) and \(q\) are prime. Directory date: | Number theory | Open | Submitted · 2mo |
[#P4] Four exponentials conjectureRead statementLet \(x_1,x_2\in\mathbb{C}\) be linearly independent over \(\mathbb{Q}\), and let \(y_1,y_2\in\mathbb{C}\) be linearly independent over \(\mathbb{Q}\). At least one of the four numbers \(e^{x_i y_j}\), with \(i,j\in\{1,2\}\), is transcendental. Directory date: | Number theory | Open | Submitted · 2mo |
[#P5] Erdős-Straus conjectureRead statementFor every integer \(n\ge 2\), there exist positive integers \(x,y,z\) such that \(\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\). Directory date: | Number theory | Open | Submitted · 2mo |
[#P6] Erdős-Moser conjectureRead statementThe equation \(1^k+2^k+\cdots+(m-1)^k=m^k\) has no solution in integers \(k>1\) and \(m>1\). Directory date: | Number theory | Open | Submitted · 2mo |
[#P7] Hodge conjectureRead statementLet \(X\) be a smooth projective complex variety and let \(p\ge 0\). Every class in \(H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)\) is a rational linear combination of cohomology classes of algebraic cycles of codimension \(p\). Directory date: | Algebraic geometry | Open | Submitted · 2mo |
[#P8] Fermat-Catalan conjectureRead statementThere are only finitely many primitive solutions \((a,b,c,m,n,k)\in\mathbb{Z}_{>0}^6\) to \(a^m+b^n=c^k\) with \(m,n,k\ge2\) and \(1/m+1/n+1/k<1\). Directory date: | Number theory | Open | Submitted · 2mo |
[#P9] Hilbert's sixteenth problem, second partRead statementFor every positive integer \(n\), there exists a finite number \(H(n)\) such that each planar polynomial vector field of degree at most \(n\) has at most \(H(n)\) limit cycles. Directory date: | Dynamical systems | Open | Submitted · 2mo |
[#P10] Sum of three cubes problemRead statementThe Diophantine equation \(x^3+y^3+z^3=k\) is solvable in integers \(x,y,z\) for each \(k\in\mathbb{Z}\) satisfying \(k\not\equiv\pm4\pmod 9\). Directory date: | Number theory | Open | Submitted · 2mo |
[#P11] Littlewood conjectureRead statementFor every \(\alpha,\beta\in\mathbb{R}\), one has \(\liminf_{n\to\infty} n\lVert n\alpha\rVert\lVert n\beta\rVert=0\), where \(\lVert x\rVert\) is the distance from \(x\) to the nearest integer. Directory date: | Diophantine approximation | Open | Submitted · 2mo |
[#P12] Falconer distance conjectureRead statementLet \(d\ge2\) and let \(E\subset\mathbb{R}^d\) be compact with \(\dim_{\mathrm H}(E)>d/2\). Then the distance set \(\{|x-y|:x,y\in E\}\) has positive one-dimensional Lebesgue measure. Directory date: | Geometric measure theory | Open | Submitted · 2mo |
[#P13] Square peg problemRead statementFor every simple closed curve \(C\subset\mathbb{R}^2\), there are four distinct points of \(C\) that are the vertices of a square. Directory date: | Geometry | Open | Submitted · 2mo |
[#P14] Perfect cuboid problemRead statementA perfect cuboid exists: there are positive integers \(a,b,c\) for which the four numbers \(\sqrt{a^2+b^2}\), \(\sqrt{a^2+c^2}\), \(\sqrt{b^2+c^2}\), and \(\sqrt{a^2+b^2+c^2}\) are integers. Directory date: | Number theory | Open | Submitted · 2mo |
[#P15] Schanuel's conjectureRead statementIf \(z_1,\ldots,z_n\in\mathbb{C}\) are linearly independent over \(\mathbb{Q}\), then \(\operatorname{trdeg}_{\mathbb{Q}}\mathbb{Q}(z_1,\ldots,z_n,e^{z_1},\ldots,e^{z_n})\ge n\). Directory date: | Transcendental number theory | Open | Submitted · 2mo |
[#P16] Birch and Swinnerton-Dyer rank conjectureRead statementFor every elliptic curve \(E/\mathbb{Q}\), one has \(\operatorname{rank}E(\mathbb{Q})=\operatorname{ord}_{s=1}L(E,s)\). Directory date: | Arithmetic geometry | Open | Submitted · 2mo |
[#P17] Navier-Stokes existence and smoothnessRead statementLet \(u_0:\mathbb{R}^3\to\mathbb{R}^3\) be smooth, divergence-free, and rapidly decaying. For every viscosity \(\nu>0\), the three-dimensional incompressible Navier-Stokes equations with initial data \(u_0\) have a globally defined smooth solution satisfying the standard energy bounds. Directory date: | Partial differential equations | Open | Submitted · 2mo |
[#P18] Artin's primitive root conjectureRead statementIf \(a\in\mathbb{Z}\) is neither \(-1\) nor a perfect square, then \(a\) is a primitive root modulo \(p\) for a positive proportion of primes \(p\). Directory date: | Number theory | Open | Submitted · 2mo |
[#P19] Collatz conjectureRead statementDefine \(T:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}\) by \(T(n)=n/2\) when \(n\) is even and \(T(n)=3n+1\) when \(n\) is odd. For every \(n\ge 1\), there exists \(k\ge 0\) such that \(T^k(n)=1\). Directory date: | Discrete dynamical systems | Open | Submitted · 2mo |
[#P20] Beal conjectureRead statementWhenever \(A^x+B^y=C^z\) for positive integers \(A,B,C\) and exponents \(x,y,z>2\), the bases have a common prime divisor, so \(\gcd(A,B,C)>1\). Directory date: | Diophantine equations | Open | Submitted · 2mo |
[#P21] Bunyakovsky conjectureRead statementLet \(f\in\mathbb{Z}[x]\) be irreducible, with positive leading coefficient and no fixed prime divisor. Then \(f(n)\) is prime for infinitely many positive integers \(n\). Directory date: | Number theory | Open | Submitted · 2mo |
[#P22] Hadamard matrix conjectureRead statementFor every positive integer \(k\), there exists a matrix \(H\in\{\pm1\}^{4k\times4k}\) such that \(HH^{\mathsf T}=4kI_{4k}\). Directory date: | Linear algebra | Open | Submitted · 2mo |
[#P23] Hadwiger's conjectureRead statementEvery finite graph \(G\) with chromatic number \(\chi(G)=k\) contains the complete graph \(K_k\) as a minor. Directory date: | Graph theory | Open | Submitted · 2mo |
[#P24] Plane Jacobian conjectureRead statementLet \(F:\mathbb{C}^2\to\mathbb{C}^2\) be a polynomial map. If the Jacobian determinant \(\det J_F\) is a nonzero constant, then \(F\) has a polynomial inverse. Directory date: | Algebraic geometry | Open | Submitted · 2mo |
[#P25] Legendre's conjectureRead statementFor every \(n\in\mathbb{Z}_{>0}\), the intersection \(\mathbb{P}\cap(n^2,(n+1)^2)\) is nonempty. Directory date: | Number theory | Open | Submitted · 2mo |
[#P26] Smooth four-dimensional Poincaré conjectureRead statementEvery smooth closed four-manifold \(M\) that is homotopy equivalent to \(S^4\) is diffeomorphic to the standard smooth four-sphere \(S^4\). Directory date: | Topology | Open | Submitted · 2mo |
[#P27] Hilbert's tenth problem over the rationalsRead statementDoes there exist an algorithm which, given any polynomial \(f\in\mathbb{Z}[x_1,\ldots,x_n]\), decides whether the equation \(f(x_1,\ldots,x_n)=0\) has a solution in \(\mathbb{Q}^n\)? Directory date: | Logic | Open | Submitted · 2mo |
[#P28] Twin prime conjectureRead statementThere are infinitely many primes \(p\) for which \(p+2\) is also prime. Directory date: | Number theory | Open | Submitted · 2mo |
[#P29] Three-dimensional Euler regularityRead statementEvery smooth divergence-free initial velocity field \(u_0:\mathbb{R}^3\to\mathbb{R}^3\) with suitable decay gives a smooth solution of the incompressible Euler equations for all \(t\ge0\). Directory date: | Partial differential equations | Open | Submitted · 2mo |
[#P30] Kakeya conjecture in dimensions at least fourRead statementLet \(d\ge4\). Every set \(E\subset\mathbb{R}^d\) that contains a unit line segment in every direction has Hausdorff dimension \(d\). Directory date: | Harmonic analysis | Open | Submitted · 2mo |
[#P31] Riemann hypothesisRead statementEvery nontrivial zero \(\rho\) of the analytically continued Riemann zeta function satisfies \(\operatorname{Re}(\rho)=\tfrac{1}{2}\). Directory date: | Analytic number theory | Open | Submitted · 2mo |
[#P32] Cramér's prime-gap conjectureRead statementIf \(p_n\) denotes the \(n\)-th prime, then \(p_{n+1}-p_n=O((\log p_n)^2)\) as \(n\to\infty\). Directory date: | Analytic number theory | Open | Submitted · 2mo |
[#P33] Lonely runner conjectureRead statementLet \(n\ge 2\) runners move around \(\mathbb{R}/\mathbb{Z}\) at distinct constant speeds \(v_1,\ldots,v_n\). For each \(i\), there exists a time \(t\) such that \(\lVert t(v_j-v_i)\rVert\ge 1/n\) for every \(j\ne i\), where \(\lVert x\rVert\) is the distance from \(x\) to the nearest integer. Directory date: | Diophantine approximation | Open | Submitted · 2mo |
[#P34] Lehmer's Mahler measure problemRead statementThere exists \(\varepsilon>0\) such that every monic polynomial \(P\in\mathbb{Z}[x]\) whose Mahler measure satisfies \(M(P)>1\) also satisfies \(M(P)\ge1+\varepsilon\). Directory date: | Number theory | Open | Submitted · 2mo |
[#P35] Infinitely many Mersenne primesRead statementInfinitely many exponents \(p\in\mathbb{P}\) have the property that \(2^p-1\in\mathbb{P}\), where \(\mathbb{P}\) is the set of prime numbers. Directory date: | Number theory | Open | Submitted · 2mo |
[#P36] Yang-Mills existence and mass gapRead statementFor every compact simple gauge group \(G\), construct a nontrivial quantum Yang-Mills theory on \(\mathbb{R}^4\) satisfying the required quantum field theory axioms and prove that its mass gap \(\Delta\) satisfies \(\Delta>0\). Directory date: | Mathematical physics | Open | Submitted · 2mo |
[#P37] Inverse Galois problemRead statementFor every finite group \(G\), there exists a finite Galois extension \(K/\mathbb{Q}\) such that \(\operatorname{Gal}(K/\mathbb{Q})\cong G\). Directory date: | Algebra | Open | Submitted · 2mo |
[#P38] Nonexistence of odd perfect numbersRead statementThere is no odd integer \(N>0\) whose sum of positive divisors satisfies \(\sigma(N)=2N\). Directory date: | Number theory | Open | Submitted · 2mo |
[#P39] P versus NPRead statementThe complexity classes \(\mathsf{P}\) and \(\mathsf{NP}\) are equal. Equivalently, every decision problem with polynomial-time verifiable yes-certificates has a deterministic polynomial-time decision algorithm. Directory date: | Computational complexity | Open | Submitted · 2mo |
[#P40] Graceful tree conjectureRead statementEvery tree \(T\) with \(m\) edges has an injective labeling \(\ell:V(T)\to\{0,1,\ldots,m\}\) for which the edge labels \(\{|\ell(u)-\ell(v)|:uv\in E(T)\}\) are exactly \(\{1,2,\ldots,m\}\). Directory date: | Graph theory | Open | Submitted · 2mo |
[#P41] Union-closed sets conjectureRead statementIf \(\mathcal{F}\) is a finite nonempty family of finite sets satisfying \(A\cup B\in\mathcal{F}\) for all \(A,B\in\mathcal{F}\), then some element belongs to at least \(|\mathcal{F}|/2\) members of \(\mathcal{F}\). Directory date: | Combinatorics | Open | Submitted · 2mo |
[#P42] Quantum PCP conjectureRead statementThere exists \(\varepsilon>0\) such that approximating the ground-state energy of a local Hamiltonian to additive error \(\varepsilon\) times the number of local terms is QMA-hard. Directory date: | Quantum computing | Open | Submitted · 2mo |
[#P43] Erdős-Hajnal conjectureRead statementFor every finite graph \(H\), there exists \(c(H)>0\) such that every \(H\)-free graph on \(n\) vertices contains a clique or an independent set of size at least \(n^{c(H)}\), where \(H\)-free means having no induced copy of \(H\). Directory date: | Graph theory | Open | Submitted · 2mo |
[#P44] Unique Games conjectureRead statementFor every \(\varepsilon,\delta>0\), there exists an alphabet size \(q\) such that it is NP-hard to distinguish unique games with optimum at least \(1-\varepsilon\) from those with optimum at most \(\delta\). Directory date: | Computational complexity | Open | Submitted · 2mo |
[#P45] Rota's basis conjectureRead statementLet \(V\) be an \(n\)-dimensional vector space and let \(B_1,\ldots,B_n\) be pairwise disjoint bases of \(V\). Their union can be partitioned into \(n\) bases, each containing exactly one vector from every \(B_i\). Directory date: | Linear algebra | Open | Submitted · 2mo |
[#P46] Zauner's conjectureRead statementFor every integer \(d\ge2\), there exist \(d^2\) unit vectors in \(\mathbb{C}^d\) whose pairwise squared inner-product magnitudes are all \(1/(d+1)\). Directory date: | Quantum information | Open | Submitted · 2mo |
[#P47] Berge-Fulkerson conjectureRead statementEvery bridgeless cubic graph \(G\) has six perfect matchings \(M_1,\ldots,M_6\) such that each edge of \(G\) belongs to exactly two of the matchings. Directory date: | Graph theory | Open | Submitted · 2mo |
[#P48] Andrews-Curtis conjectureRead statementEvery balanced presentation \(\langle x_1,\ldots,x_n\mid r_1,\ldots,r_n\rangle\) of the trivial group can be transformed into \(\langle x_1,\ldots,x_n\mid x_1,\ldots,x_n\rangle\) by Andrews-Curtis moves. Directory date: | Group theory | Open | Submitted · 2mo |
[#P49] Graph reconstruction conjectureRead statementEvery finite simple graph \(G\) with at least three vertices is determined up to isomorphism by the multiset \(\{G-v:v\in V(G)\}\) of its one-vertex-deleted subgraphs. Directory date: | Graph theory | Open | Submitted · 2mo |
[#P50] Kaplansky zero-divisor conjectureRead statementIf \(K\) is a field and \(G\) is a torsion-free group, then the group algebra \(K[G]\) has no nonzero zero divisors. Directory date: | Algebra | Open | Submitted · 2mo |
[#P51] Novikov conjectureRead statementFor every discrete group \(\Gamma\), the higher signatures associated with classes in \(H^*(B\Gamma;\mathbb{Q})\) are invariant under oriented homotopy equivalences of closed manifolds. Directory date: | Topology | Open | Submitted · 2mo |
[#P52] Hadwiger-Nelson problemRead statementDetermine the chromatic number \(\chi(\mathbb{R}^2)\) of the unit-distance graph on the Euclidean plane, whose vertices are points of \(\mathbb{R}^2\) and whose edges join pairs at distance \(1\). Directory date: | Discrete geometry | Open | Submitted · 2mo |
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