Mathematics Problem Archive

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....

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 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 $\varepsilon>0$, 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 $X_t$ be two-dimensional Brownian motion. (i) For every pair $x,y \notin X[0,1]$, is there a Jordan arc $\Gamma$ containing $x$ and $y$ such that ...

For a set $B$, define its percolation dimension as the infimum of the Hausdorff dimensions of Jordan arcs $A\subset B$ containing at least two distinc...

Let $D$ be a triangle whose angles are all strictly less than $\pi/2$, and let $\mu_2>0$ be the second eigenvalue of the Laplacian on $D$ with Neumann...

Let $D\subset\mathbb{R}^2$ be a connected open set with smooth boundary, and let $X,Y$ be synchronously coupled reflected Brownian motions in $D$ driv...

For each $k\in\mathbb{Z}$, let $B^k$ be Brownian motion and $T_k$ a stopping time, with the stopped pieces independent, $0\le T_k<\infty$, and with th...

Choose uniformly a bijective labeling of the vertices of the $n\times n$ discrete square by $1,2,\ldots,n^2$, and call a vertex a peak when all adjace...

On $\mathbb{Z}^2$, take nearest-neighbor jump probabilities $p_1,q_1,p_2,q_2$ in directions $\pm e_1,\pm e_2$, with $p_1>q_1$ and $p_2>q_2$. If the an...

Give a mathematically useful definition of a process being a 'martingale for practical purposes', so that failure means it is practical to find a stop...