Unsolved Problems

If $B$ is an additive basis of order 2, must the representation function tend to infinity?...

If the sum of $m$ $k$-th powers equals the sum of $n$ $k$-th powers, must $m + n \geq k$?...

Can every odd integer greater than 5 be expressed as the sum of an odd prime and an even semiprime?...

Does every nonnegative integer appear in Recamán's sequence?...

Can an algorithm determine if a constant-recursive sequence contains a zero?...

What are the exact values of $g(k)$ and $G(k)$ for all $k$ in Waring's problem?...

Do the Ulam numbers have a positive density?...

Are there infinitely many real quadratic number fields with unique factorization?...

Can the Kronecker-Weber theorem on abelian extensions of $\mathbb{Q}$ be extended to any base number field?...

Does the $p$-adic regulator of an algebraic number field not vanish?...

For all $\varepsilon > 0$, does $\zeta(1/2 + it) = o(t^\varepsilon)$ as $t \to \infty$?...

Do the nontrivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator?...

Do all automorphic L-functions have their nontrivial zeros on the critical line?...

Does the pair correlation function of Riemann zeta zeros match that of random Hermitian matrices?...

What is the optimal exponent in the error term for the divisor summatory function?...

Is there a dense set of points in the plane with all pairwise distances rational?...

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...

Is the Euler-Mascheroni constant $\gamma$ irrational?...

Is $\zeta(3) = 1 + 1/8 + 1/27 + 1/64 + \cdots$ transcendental?...

For any two real numbers $\alpha, \beta$, does $\liminf_{n \to \infty} n \|n\alpha\| \|n\beta\| = 0$?...

Can integer factorization be solved in polynomial time on a classical computer?...

For $A^x + B^y = C^z$ with $x, y, z > 2$, must $A$, $B$, and $C$ share a common prime factor?...

Are there finitely many solutions to $a^m + b^n = c^k$ with coprime $a,b,c$ and $1/m + 1/n + 1/k < 1$?...

Does an irreducible integer polynomial with no fixed prime divisor produce infinitely many primes?...

Do finitely many linear forms simultaneously take prime values infinitely often, barring congruence obstructions?...

Are there always at least 4 primes between consecutive squares of primes $p_n^2$ and $p_{n+1}^2$?...

Is $p$ prime if and only if $pB_{p-1} \equiv -1 \pmod{p}$ for the Bernoulli number $B_{p-1}$?...

Do primes distribute uniformly in arithmetic progressions up to nearly $x$ (instead of $x^{1/2}$)?...

Is there a relation between the order of the center of the Steinberg group and the Dedekind zeta function?...

Is $\|f(A)\| \leq 2\sup_{z \in W(A)} |f(z)|$ for all matrices $A$ and functions $f$ analytic on the numerical range?...

Does there exist a rectangular cuboid with integer edges, face diagonals, and space diagonal?...

Do symmetric informationally complete POVMs exist in all dimensions?...

Can every balanced presentation of the trivial group be transformed to a trivial presentation by Nielsen moves?...

Can a finite system of left cosets forming a partition of a group have distinct indices?...

Is there a minimum positive Mahler measure for non-cyclotomic polynomials?...

For which domains do non-zero functions exist with zero integrals over all congruent copies?...

Do smooth initial data for 3D Navier-Stokes equations yield smooth solutions for all time?...

Does every non-totally-ordered finite poset have two elements with probability between 1/3 and 2/3 in random linear extensions?...

If $k$ runners with distinct speeds run on a circular track, will each be lonely (distance $\geq 1/k$ from others) at some time?...

For a finite family of sets closed under unions, must some element appear in at least half the sets?...

What is the maximum number of points in an $n \times n$ grid with no three collinear?...

For fixed $r$, can the number of size-$k$ sets needed for an $r$-sunflower be bounded by $c^k$ for some constant $c$?...

Does every bridgeless graph have a collection of cycles covering each edge exactly twice?...

For any fixed graph $H$, do $H$-free graphs contain large cliques or independent sets?...

Does every finite connected vertex-transitive graph have a Hamiltonian path?...

What is the chromatic number of the plane with unit distance graph coloring?...

Can unknots be recognized in polynomial time?...

Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?...

Do quantum invariants of knots relate asymptotically to hyperbolic volume?...

Are certain combinations of Pontryagin classes homotopy invariant?...