Mathematics Problem Archive

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...

For every $\varepsilon>0$, do constants $T_0(\varepsilon),c(\varepsilon)>0$ exist such that, for $T\geq T_0$ and $H=T^{1/2+\varepsilon}$, the number $...

Let $f(x)\in\mathbb{Z}[x]$ be irreducible over $\mathbb{Q}$ with all roots real, let $f(\alpha)=0$, let $k=\mathbb{Q}(\alpha)$, and let $\mathcal{O}_k...

For $\mathcal{O}_k$ as in item 7.1, let $\Gamma_{\mathrm{unit}}$ have vertex set $\mathcal{O}_k$, joining two elements when their difference is a unit...

If $k$ is a number field that is not totally real, is there a hyperbolic $3$-manifold with trace field $k$?...