Mathematics Problem Archive

Is every positive integer expressible as a sum of at most five tetrahedral numbers?...

Let $K$ be a number field and let $\rho$ be an irreducible $p$-adic representation of $\operatorname{Gal}(\overline K/K)$ that is unramified outside f...

For the classical-group pairs and generic local $L$-parameters in the Gan–Gross–Prasad setting, is there exactly one relevant representation $\pi$ in ...

For every totally real number field $F$ and prime $p$, do the Iwasawa invariants $\lambda(F_\infty/F)$ and $\mu(F_\infty/F)$ of the cyclotomic $\mathb...

Hermite's problem: is it possible, for any natural number $n$, to assign a sequence of natural numbers to each real number such that the sequence for ...

Kummer–Vandiver conjecture: primes $p$ do not divide the class number of the maximal real subfield of the $p$-th cyclotomic field....

Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant $X$ is within a constant multiple o...

For an Artin $L$-function attached to a Galois extension of number fields, is its leading Taylor coefficient at $s=0$ the product of the corresponding...

For a motive (or the cohomology of a smooth projective variety) and an appropriate integer argument, is the order of vanishing of its L-function the p...

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....

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 $1/4$....

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...

If the base-$b$ expansion of a real number can be emitted in real time by a multitape Turing machine (bounded time between successive digits), must th...

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...

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?...