Mathematics Problem Archive

Find the value of the De Bruijn–Newman constant....

Is Selberg class of Dirichlet series equal to class of automorphic L-functions?...

For every $\varepsilon>0$, is there a $T_0(\varepsilon)$ such that, whenever $T\geq T_0$ and $H=T^{1/4+\varepsilon}$, the interval $(T,T+H]$ contains ...

For fixed admissible $k$, does $T^{-1}\int_0^T|\zeta(1/2+it)|^{2k}\,dt$ have the Keating–Snaith asymptotic $a(k)G(k+1)^2G(2k+1)^{-1}(\log T)^{k^2}$, w...

The density hypothesis for zeroes of the Riemann zeta function....

Generalized Riemann hypothesis for Selberg class: do the nontrivial zeros of all functions in Selberg class lie on the critical line $1/2 + it$ with r...

Piltz divisor problem on bounding $\Delta_k(x) = D_k(x) - xP_k(\log(x))$...

Let $K$ be a number field and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$ with unitary central character. Is ev...

Selberg's 1/4 conjecture: the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least $1/4$....

Selberg's orthogonality conjecture: generalization of Mertens' theorem for functions in Selberg class....

Bombieri–Lang conjecture: $K$-rational points on a variety of general type over a number field $K$ are not a dense set in Zariski topology....

Manin conjecture: if K-rational points on Fano variety are Zariski-dense subset, then the distribution of points of height: $H(x)\leq B$ in any Zarisk...

For an abelian variety or suitable motive over a number field, are its normalized Frobenius conjugacy classes equidistributed in the associated compac...

Vojta's conjecture: points on non-singular algebraic variety over algebraic number field that not satisfy certain height inequality are contained in s...

Fix $n\geq3$. If coprime nonzero integers $a_1,\ldots,a_n$ have sum zero and no proper subsum zero, is it true that for every $\varepsilon>0$ there is...

Szpiro's conjecture: for any $\varepsilon > 0$, there is some constant $C(\varepsilon)$ such that, for any elliptic curve $E$ defined over $\mathbb{Q}...

Zilber–Pink conjecture that if $X$ is a mixed Shimura variety or semiabelian variety defined over $\mathbb{C}$, and $V \subseteq X$ is a subvariety, t...

Can a discrete logarithm on a elliptic curve be computed in sub-exponential time?...

Does every rational number with an odd denominator have an odd greedy expansion?...

Which transcendental numbers are (exponential) periods?...

How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of specific (suspected) transcendental numbers such a...

Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem): determine precisely what rational numbers are c...

Erdős–Moser problem: is $1^1 + 2^1 = 3^1$ the only solution to the Erdős–Moser equation?...

Goormaghtigh conjecture on solutions to $(x^m - 1)/(x - 1) = (y^n - 1)/(y - 1)$ where $x > y > 1$ and $m, n > 2$....

The uniqueness conjecture for Markov numbers that every Markov number is the largest number in exactly one normalized solution to the Markov Diophanti...

Pillai's conjecture: for any $A, B, C$, the equation $Ax^m - By^n = C$ has finitely many solutions when $m, n$ are not both $2$....

Which integers can be written as the sum of three perfect cubes?...

Can every integer be written as a sum of four perfect cubes?...

Agrawal's conjecture that given coprime positive integers $n$ and $r$, if $(X - 1)^n \equiv X^n - 1 \pmod{n, X^r - 1}$, then either $n$ is prime or $n...

Dubner's conjecture: every even number greater than $4208$ is the sum of two primes which both have a twin....

Erdős–Mollin–Walsh conjecture: no three consecutive numbers are all powerful....

Feit–Thompson conjecture: for all distinct prime numbers $p$ and $q$, $(p^q - 1)/(p - 1)$ does not divide $(q^p - 1)/(q - 1)$...

Fortune's conjecture that no Fortunate number is composite....

Gillies' conjecture on the distribution of prime divisors of Mersenne numbers....

For coprime integers $1\leq a<d$, is the least prime $p(a,d)$ congruent to $a\pmod d$ always less than $d^2$?...

New Mersenne conjecture: for any odd natural number $p$, if any two of the three conditions $p = 2^k \pm 1$ or $p = 4^k \pm 3$, $2^p - 1$ is prime, an...

Selfridge's conjecture: is 78,557 the lowest Sierpiński number?...

Does the converse of Wolstenholme's theorem hold for all natural numbers?...

Are all Euclid numbers square-free?...

Are there any composite c satisfying 2^{c − 1} ≡ 1 (mod c^{2})?...

Are there any Wieferich primes in base 47?...

Are there infinitely many balanced primes?...

Are there infinitely many cluster primes?...

Are there infinitely many cousin primes?...

Are there infinitely many Cullen primes?...

Are there infinitely many Euclid primes?...

Are there infinitely many Fibonacci primes?...

Are there infinitely many Kummer primes?...

Are there infinitely many Kynea primes?...

Are there infinitely many Lucas primes?...