Mathematics Problem Archive

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?...

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?...

Are there infinitely many Newman–Shanks–Williams primes?...

Are there infinitely many palindromic primes to every base?...

Are there infinitely many Pell primes?...

Are there infinitely many Pierpont primes?...

Are there infinitely many prime quadruplets?...

Are there infinitely many prime triplets?...

Siegel's conjecture: are there infinitely many regular primes, and if so is their natural density as a subset of all primes $e^{-1/2}$?...

Are there infinitely many sexy primes?...

Are there infinitely many Wagstaff primes?...

Are there infinitely many Wieferich primes?...

Are there infinitely many Wilson primes?...

Are there infinitely many Wolstenholme primes?...

Are there infinitely many Woodall primes?...

Can a prime p satisfy $2^{p-1}\equiv 1\pmod{p^2}$ and $3^{p-1}\equiv 1\pmod{p^2}$ simultaneously?...

Does every prime number appear in the Euclid–Mullin sequence?...

What is the smallest Skewes's number?...

For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fi...

For any given integer a > 0, are there infinitely many primes p such that a^{p − 1} ≡ 1 (mod p^{2})?...

For any given integer b which is not a perfect power and not of the form −4k^{4} for integer k, are there infinitely many repunit primes to base b?...

For any given integers $k\geq 1, b\geq 2, c\neq 0$, with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form $(k\times b^n+c...

Is every Fermat number $2^{2^n} + 1$ composite for $n > 4$?...

Is 509,203 the lowest Riesel number?...

Is every positive integer expressible as a sum of at most seven octahedral numbers?...

Let $F$ be totally real, let $\widetilde F$ be the compositum of all $\mathbb Z_p$-extensions of $F$, let $\widetilde L$ be its maximal unramified abe...

For every number field $K$ and every prime $\ell$, does the Iwasawa invariant $\mu_\ell(K)$ vanish?...

For every odd prime $p$ and every positive integer $t$, does there exist a $p$-rational number field $K$ with $\operatorname{Gal}(K/\mathbb Q)\cong(\m...

For a finite abelian extension of number fields, does the Brumer–Stark element have the predicted ideal-class annihilation and principalization proper...