Mathematics Problem Archive

Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating....

Are there any pairs of betrothed numbers which have same parity?...

Are there any pairs of relatively prime amicable numbers?...

Are there infinitely many betrothed numbers?...

Do any odd noncototients exist?...

Do any (2, 5)-perfect numbers exist?...

Do any Taxicab(5, 2, n) exist for n > 1?...

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

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

Characterize all algebraic number fields that have some power basis....

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

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...

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

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...

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 $2$....

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....

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?...