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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For which number fields does Hilbert's tenth problem hold?
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Kueker's conjecture
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The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for \aleph_1-saturated models of a countable theory.
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Shelah's categoricity conjecture for L_{\omega_1,\omega}
If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.
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Shelah's eventual categoricity conjecture
For every cardinal \lambda there exists a cardinal \mu(\lambda) such that if an AEC K with LS(K){} \le \lambda is categorical in a cardinal above \mu(\lambda) then it is categorical in all cardinals above \mu(\lambda).
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The stable field conjecture
every infinite field with a stable first-order theory is separably closed.
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The stable forking conjecture for simple theories
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Tarski's exponential function problem
is the theory of the real numbers with the exponential function decidable?
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The universality problem for C-free graphs
For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?
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The universality spectrum problem
Is there a first-order theory whose universality spectrum is minimum?
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Vaught conjecture
the number of countable models of a first-order complete theory in a countable language is either finite, \aleph_0, or 2^{\aleph_0}.
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Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality…
Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality \aleph_{\omega_1} does it have a model of cardinality continuum?
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Do the Henson graphs have the finite model property?
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Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
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Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
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If the class of atomic models of a complete first order theory is categorical in the \aleph_n, is it categorical in every cardinal?
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Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the…
Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
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Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?
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Is the theory of the field of Laurent series over \mathbb{Z}_p decidable? of the field of polynomials over \mathbb{C}?
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Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?
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Determine the structure of Keisler's order.
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What is the nature of the proof-theoretic ordinal (the smallest ordinal a theory cannot prove well-founded) for second-order arithmetic,…
What is the nature of the proof-theoretic ordinal (the smallest ordinal a theory cannot prove well-founded) for second-order arithmetic, ZFC, or stronger theories?
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Ibragimov–Iosifescu conjecture for φ-mixing sequences
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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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Data last refreshed 2026-10-20 03:01 UTC.