The world's unsolved problems
993 open questions across 14 fields, gathered from Wikipedia. Pick one and start thinking.
488 results in Mathematics · clear · surprise me
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Hartmanis–Stearns conjecture
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Beal's conjecture
for all integral solutions to A^x + B^y = C^z where x, y, z > 2, all three numbers A, B, C must share some prime factor.
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Brocard's problem
are there any integer solutions to n! + 1 = m^{2} other than n = 4, 5, 7?
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Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
determine precisely what rational numbers are congruent numbers.
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Erdős–Moser problem
is 1^1 + 2^1 = 3^1 the only solution to the Erdős–Moser equation?
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Erdős–Straus conjecture
for every n \geq 2, there are positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z.
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Fermat–Catalan conjecture
there are finitely many distinct solutions (a^m, b^n, c^k) to the equation a^m + b^n = c^k with a, b, c being positive coprime integers and m, n, k being positive integers satisfying 1/m + 1/n + 1/k < 1.
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Goormaghtigh conjecture on solutions to (x^m
1)/(x - 1) = (y^n - 1)/(y - 1) where x > y > 1 and m, n > 2.
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The uniqueness conjecture for Markov numbers
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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.
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Which integers can be written as the sum of three perfect cubes?
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Can every integer be written as a sum of four perfect cubes?
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Agoh–Giuga conjecture on the Bernoulli numbers that p is prime if and only if pB_{p-1} \equiv -1 \pmod p
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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^{2} \equiv 1 \pmod{r}
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Artin's conjecture on primitive roots that if an integer is neither a perfect square nor -1, then it is a primitive root modulo infinitely…
Artin's conjecture on primitive roots that if an integer is neither a perfect square nor -1, then it is a primitive root modulo infinitely many prime numbers p
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Brocard's conjecture
there are always at least 4 prime numbers between consecutive squares of prime numbers, aside from 2^{2} and 3^{2}.
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Bunyakovsky conjecture
if an integer-coefficient polynomial f has a positive leading coefficient, is irreducible over the integers, and has no common factors over all f(x) where x is a positive integer, then f(x) is prime infinitely often.
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Catalan's Mersenne conjecture
some Catalan–Mersenne number is composite and thus all Catalan–Mersenne numbers are composite after some point.
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Dickson's conjecture
for a finite set of linear forms a_1 + b_1 n, \ldots, a_k + b_k n with each b_i \geq 1, there are infinitely many n for which all forms are prime, unless there is some congruence condition preventing it.
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Dubner's conjecture
every even number greater than 4208 is the sum of two primes which both have a twin.
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Elliott–Halberstam conjecture on the distribution of prime numbers in arithmetic progressions.
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Erdős–Mollin–Walsh conjecture
no three consecutive numbers are all powerful.
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Feit–Thompson conjecture
for all distinct prime numbers p and q, (p^q - 1)/(p - 1) does not divide (q^p - 1)/(q - 1)
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Fortune's conjecture that no Fortunate number is composite.
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The Gaussian moat problem
is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
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Gillies' conjecture on the distribution of prime divisors of Mersenne numbers.
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Landau's problems
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Problems associated to Linnik's theorem
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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, and (2^{p} + 1)/3 is prime are true, then the third condition is also true.
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Polignac's conjecture
for all positive even numbers n, there are infinitely many prime gaps of size n.
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Data last refreshed 2026-10-20 03:01 UTC.