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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Büchi's problem on sufficiently large sequences of square numbers with constant second difference.
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Carmichael's totient function conjecture
do all values of Euler's totient function have multiplicity greater than 1?
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Catalan–Dickson conjecture on aliquot sequences
no aliquot sequences are infinite but non-repeating.
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Exponent pair conjecture
for all \varepsilon > 0, is the pair (\varepsilon, 1/2 + \varepsilon) an exponent pair?
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The Gauss circle problem
how far can the number of integer points in a circle centered at the origin be from the area of the circle?
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Grimm's conjecture
each element of a set of consecutive composite numbers can be assigned a distinct prime number that divides it.
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Hall's conjecture
for any \varepsilon > 0, there is some constant c(\varepsilon) such that either y^2 = x^3 or |y^2 - x^3| > c(\varepsilon)x^{1/2 - \varepsilon}.
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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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Data last refreshed 2026-10-20 03:01 UTC.