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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Are there infinitely many prime quadruplets?
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Are there infinitely many prime triplets?
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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}?
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Are there infinitely many sexy primes?
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Are there infinitely many safe and Sophie Germain primes?
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Are there infinitely many Wagstaff primes?
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Are there infinitely many Wieferich primes?
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Are there infinitely many Wilson primes?
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Are there infinitely many Wolstenholme primes?
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Are there infinitely many Woodall primes?
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Can a prime p satisfy 2^{p-1}\equiv 1\pmod{p^2} and 3^{p-1}\equiv 1\pmod{p^2} simultaneously?
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Does every prime number appear in the Euclid–Mullin sequence?
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What is the smallest Skewes's number?
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For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1,…
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 Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
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For any given integer a > 0, are there infinitely many primes p such that a'p − 1 ≡ 1 (mod p2)?
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For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to…
For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b?
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For any given integers k\geq 1, b\geq 2, c\neq 0, with 1=gcd(k, c) = 1 and 1=gcd(b, c) = 1, are there infinitely many primes of the form…
For any given integers k\geq 1, b\geq 2, c\neq 0, with 1=gcd(k, c) = 1 and 1=gcd(b, c) = 1, are there infinitely many primes of the form (k\times b^n+c)/\gcd(k+c,b-1) with integer n ≥ 1?
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Is every Fermat number 2^{2^n} + 1 composite for n > 4?
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Is 509,203 the lowest Riesel number?
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Does the partition principle (PP) imply the axiom of choice (AC)?
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(Woodin) Does the generalized continuum hypothesis below a strongly compact cardinal imply the generalized continuum hypothesis everywhere?
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Does the generalized continuum hypothesis entail {\diamondsuit(E^{\lambda^+}_{\operatorname{cf}(\lambda)}}) for every singular cardinal…
Does the generalized continuum hypothesis entail {\diamondsuit(E^{\lambda^+}_{\operatorname{cf}(\lambda)}}) for every singular cardinal \lambda?
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Does the generalized continuum hypothesis imply the existence of an ℵ2-Suslin tree?
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If ℵω is a strong limit cardinal, is 2^{\aleph_\omega} ω4, was obtained by Shelah using his PCF theory.
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The problem of finding the ultimate core model, one that contains all large cardinals.
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Woodin's Ω-conjecture
if there is a proper class of Woodin cardinals, then Ω-logic satisfies an analogue of Gödel's completeness theorem.
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Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
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Does there exist a Jónsson algebra on ℵω?
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Is OCA (the open coloring axiom) consistent with 2^{\aleph_{0}}>\aleph_{2}?
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Reinhardt cardinals
Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
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Data last refreshed 2026-10-20 03:01 UTC.