The world's unsolved problems
993 open questions across 14 fields, gathered from Wikipedia. Pick one and start thinking.
Showing all 993 problems · surprise me
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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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Schinzel's hypothesis H that for every finite collection \{f_1, \ldots, f_k\} of nonconstant irreducible polynomials over the integers with…
Schinzel's hypothesis H that for every finite collection \{f_1, \ldots, f_k\} of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers n for which f_1(n), \ldots, f_k(n)...
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Selfridge's conjecture
is 78,557 the lowest Sierpiński number?
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Does the converse of Wolstenholme's theorem hold for all natural numbers?
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Are all Euclid numbers square-free?
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Are all Fermat numbers square-free?
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Are all Mersenne numbers of prime index square-free?
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Are there any composite c satisfying 2c − 1 ≡ 1 (mod c2)?
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Are there any Wall–Sun–Sun primes?
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Are there any Wieferich primes in base 47?
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Are there infinitely many balanced primes?
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Are there infinitely many cluster primes?
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Are there infinitely many cousin primes?
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Are there infinitely many Cullen primes?
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Are there infinitely many Euclid primes?
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Are there infinitely many Fibonacci primes?
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Are there infinitely many Kummer primes?
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Are there infinitely many Kynea primes?
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Are there infinitely many Lucas primes?
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Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
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Are there infinitely many Newman–Shanks–Williams primes?
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Are there infinitely many palindromic primes to every base?
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Are there infinitely many Pell primes?
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Are there infinitely many Pierpont primes?
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Data last refreshed 2026-10-20 03:01 UTC.