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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Is there a covering system with odd distinct moduli?
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Is \pi a normal number (i.e., is each digit 0–9 equally frequent)?
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Are all irrational algebraic numbers normal?
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Is 10 a solitary number?
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Erdős conjecture on arithmetic progressions that if the sum of the reciprocals of the members of a set of positive integers diverges, then…
Erdős conjecture on arithmetic progressions that if the sum of the reciprocals of the members of a set of positive integers diverges, then the set contains arbitrarily long arithmetic progressions.
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Erdős–Turán conjecture on additive bases
if B is an additive basis of order 2, then the number of ways that positive integers n can be expressed as the sum of two numbers in B must tend to infinity as n tends to infinity.
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Gilbreath's conjecture on consecutive applications of the unsigned forward difference operator to the sequence of prime numbers.
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Goldbach's conjecture
every even natural number greater than 2 is the sum of two prime numbers.
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Lander, Parkin, and Selfridge conjecture
if the sum of m k-th powers of positive integers is equal to a different sum of n k-th powers of positive integers, then m + n \geq k.
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Lemoine's conjecture
all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime.
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Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two…
Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set \{1, \ldots, 2n\}
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Pollock's conjectures
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Does every nonnegative integer appear in Recamán's sequence?
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Skolem problem
can an algorithm determine if a constant-recursive sequence contains a zero?
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The values of g(k) and G(k) in Waring's problem
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Do the Ulam numbers have a positive density?
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Determine growth rate of r'k(N) (see Szemerédi's theorem)
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Class number problem
are there infinitely many real quadratic number fields with unique factorization?
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Fontaine–Mazur conjecture
actually numerous conjectures, all proposed by Jean-Marc Fontaine and Barry Mazur.
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Gan–Gross–Prasad conjecture
a restriction problem in representation theory of real or p-adic Lie groups.
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Greenberg's conjectures
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Hermite's problem
is it possible, for any natural number n, to assign a sequence of natural numbers to each real number such that the sequence for x is eventually periodic if and only if x is algebraic of degree n?
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Hilbert's 9th problem
find the most general reciprocity law for the norm residues of k-th order in a general algebraic number field, where k is a power of a prime.
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Hilbert's 12th problem
extend the Kronecker–Weber theorem on Abelian extensions of \mathbb{Q} to any base number field.
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Kummer–Vandiver conjecture
primes p do not divide the class number of the maximal real subfield of the p-th cyclotomic field.
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Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant X is within a constant…
Lang and Trotter's conjecture on supersingular primes that the number of supersingular primes less than a constant X is within a constant multiple of \sqrt{X}/\ln{X}
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Leopoldt's conjecture
a p-adic analogue of the regulator of an algebraic number field does not vanish.
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Stark conjectures (including Brumer–Stark conjecture)
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Characterize all algebraic number fields that have some power basis.
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Beilinson's conjectures about special values of L-functions.
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Data last refreshed 2026-10-20 03:01 UTC.