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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Location of nontrivial zeros of L-functions, especially conjectures concerning their critical lines
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Montgomery's pair correlation conjecture
the normalized pair correlation function between pairs of zeros of the Riemann zeta function is the same as the pair correlation function of random Hermitian matrices.
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Piltz divisor problem on bounding \Delta_k(x) = D_k(x)
xP_k(\log(x))
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Ramanujan–Petersson conjecture
a number of generalizations of the original conjecture, concerning growth rate of coefficients of automorphic forms.
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Selberg's 1/4 conjecture
the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1/4.
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Selberg's orthogonality conjecture
generalization of Mertens' theorem for functions in Selberg class.
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Bombieri–Lang conjecture
K -rational points on a variety of general type over a number field K are not a dense set in Zariski topology.
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Erdős–Ulam problem
Is there a dense set of points in the plane all at rational distances from one another?
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Manin conjecture
if K-rational points on Fano variety are Zariski-dense subset, then the distribution of points of height: H(x)\leq B in any Zariski-open subset U is proportional to B \log (B)^{r-1}, where r is rank of Picard group of that variety.
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Sato–Tate conjecture
also a number of related conjectures that are generalizations of the original conjecture.
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In Unit square
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Vojta's conjecture
points on non-singular algebraic variety over algebraic number field that not satisfy certain height inequality are contained in some Zariski-closed set.
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n conjecture
a generalization of the abc conjecture to more than three integers.
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Zilber–Pink conjecture that if X is a mixed Shimura variety or semiabelian variety defined over \mathbb{C}, and V \subseteq X is a…
Zilber–Pink conjecture that if X is a mixed Shimura variety or semiabelian variety defined over \mathbb{C}, and V \subseteq X is a subvariety, then V contains only finitely many maximal atypical subvarieties.
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Can integer factorization be done in polynomial time?
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Can a discrete logarithm be computed in polynomial time?
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Does every rational number with an odd denominator have an odd greedy expansion?
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Littlewood conjecture
for any two real numbers \alpha, \beta, \liminf_{n \rightarrow \infty} n\,\Vert n\alpha\Vert\,\Vert n\beta\Vert = 0, where \Vert x\Vert is the distance from x to the nearest integer.
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Schanuel's conjecture on the transcendence degree of certain field extensions of the rational numbers. In particular: Are \pi and e…
Schanuel's conjecture on the transcendence degree of certain field extensions of the rational numbers. In particular: Are \pi and e algebraically independent? Which nontrivial combinations of transcendental numbers (such as e + \pi, e\pi, \pi^e, \pi^{\pi},...
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The four exponentials conjecture
the transcendence of at least one of four exponentials of combinations of irrationals
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Are Euler's constant \gamma and Catalan's constant G irrational? Are they transcendental? Is Apéry's constant \zeta(3) transcendental?For…
Are Euler's constant \gamma and Catalan's constant G irrational? Are they transcendental? Is Apéry's constant \zeta(3) transcendental?For some background on the numbers in this problem, see articles by Eric W. Weisstein at Wolfram MathWorld (all articles...
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Euler's Constant
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Catalan's Constant
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Apéry's Constant
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irrational numbers ()
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transcendental numbers ()
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irrationality measures ()
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Which transcendental numbers are (exponential) periods?
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How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of specific (suspected) transcendental…
How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of specific (suspected) transcendental numbers such as \pi and \gamma?
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Which irrational numbers have simple continued fraction terms whose geometric mean converges to Khinchin's constant?
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Data last refreshed 2026-10-20 03:01 UTC.