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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Lehmer's totient problem
if \phi(n) divides n - 1, must n be prime?
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Magic square of squares
is there a 3x3 magic square composed of distinct perfect squares?
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Mahler's 3/2 problem that no real number x has the property that the fractional parts of x(3/2)^n are less than 1/2 for all positive…
Mahler's 3/2 problem that no real number x has the property that the fractional parts of x(3/2)^n are less than 1/2 for all positive integers n.
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Newman's conjecture
the partition function satisfies any arbitrary congruence infinitely often.
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Scholz conjecture
the length of the shortest addition chain producing 2^n - 1 is at most n - 1 plus the length of the shortest addition chain producing n.
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Singmaster's conjecture
is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
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Sister Beiter conjecture
the maximal coefficient of a ternary cyclotomic polynomial is bounded by 2/3 of the smallest prime factor of its index.
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Are there infinitely many perfect numbers?
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Do any odd perfect numbers exist?
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Do quasiperfect numbers exist?
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Do any non-power of 2 almost perfect numbers exist?
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Are there 65, 66, or 67 idoneal numbers?
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Are there any pairs of amicable numbers which have opposite parity?
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Are there any pairs of betrothed numbers which have same parity?
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Are there any pairs of relatively prime amicable numbers?
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Are there infinitely many pairs of amicable numbers?
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Are there infinitely many betrothed numbers?
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Are there infinitely many Giuga numbers?
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Do any Lychrel numbers exist in base 10?
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Do any odd noncototients exist?
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Do any odd weird numbers exist?
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Do any (2, 5)-perfect numbers exist?
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Do any Taxicab(5, 2, n) exist for n > 1?
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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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Data last refreshed 2026-10-20 03:01 UTC.