Kaplansky's Conjectures
The zero-divisor conjecture If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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Köthe conjecture
The Köthe conjecture: In any ring, the sum of two nil left ideals is nil.
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Kotzig's Conjecture
For any tree T with n edges, the complete graph K_2n+1 decomposes into 2n+1 edge-disjoint copies of T via cyclic shifts of a single embedding.
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Kummer–Vandiver conjecture
Kummer–Vandiver conjecture states that for every prime p, the class number of the maximal real subfield of ℚ(ζ_p) is not divisible by p. -
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Kurepa's conjecture
## Kurepa's conjecture For all n, !nnot≡ 0 mod n This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by Richard K. Guy
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Lander, Parkin, and Selfridge Conjecture
The Lander–Parkin–Selfridge conjecture: if the sum of n positive integer k-th powers equals the sum of m positive integer k-th powers, with all values on the left distinct from all values on the right, then n + m ≥ k.
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Largest odd divisor of a(n-1) + textrmprime(n)
Is a(33900) the last term equal to 1?
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Latin Tableau Conjecture
The Latin Tableau Conjecture: If G is the simple graph of a Young diagram, then G is CDS-colorable.
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Least k > 0 such that (k+1)(k+2)⋯(k+n) + 1 is prime
(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2, which is never prime. Hence a(4) = 0. Conjecture: a(n) = 0 if and only if n = 4.
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Least k ≥ 1 such that 2 · n^k - 1 is prime
Is a(n) defined for all n ≥ 2? That is, does there exist k > 0 such that 2 · n^k - 1 is prime?
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Least k such that cyclotomic polynomial Φ_k(n) is prime
Is a(n) defined for all n ≥ 1? That is, for every n ≥ 1, does there exist k > 0 such that |Φ_k(n)| is prime?
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Least m such that φ(m) = n!
Conjecture: unless n! + 1 is prime (i.e., n ∈ A002981), a(n) = p q where p is the least prime > √(n!) such that (p - 1) | n! and q = n!/p - 1 + 1 is prime. - M. F.
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Least positive multiple of n in base 10 with digits 0 and 1
It is known that a(10^k - 1) = (10^9k - 1) / 9 for all k. Is a(n) < a(10^k - 1) for all n < 10^k - 1? - David Radcliffe, Aug 01 2025
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Least prime ≥ n
According to the "k-tuple" conjecture, a(n) is the initial term of the lexicographically earliest increasing arithmetic progression of n primes; the corresponding common differences are given by A061558.
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Legendre's conjecture
Does there always exist at least one prime between consecutive perfect squares?
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Lehmer's Mahler measure problem
Let M(f) denote the Mahler measure of f. There exists a constant μ>1 such that for any f(x)∈ℤ[x], M(f)>1 → M(f)≥μ.
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Lehmer's totient problem
Does there exist a composite number n > 1 such that Euler’s totient function φ(n) divides n - 1?
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Leinster Groups
Conjecture: Are there infinitely many Leinster groups? This asks whether there exist infinitely many (non-isomorphic) finite groups that are Leinster groups. Formalized via the negation of "Does there exist an n such that all Leinster groups have order less than n".
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Lemoine's conjectures
For all odd integers n ≥ 7 there are prime numbers p,q such that n = p+2q.
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Littlewood conjectures
For any two real numbers α and β, liminf_n→∞ n‖|nα‖|‖|nβ‖| = 0 where ‖|x‖| := min(|x - ⌊ x ⌋|, |x - ⌈ x ⌉|) is the distance to the nearest integer.
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Local uniformization
Local uniformization in positive characteristic. Let k be a field of characteristic p > 0, let F be a finitely generated field extension of k, and let O be a valuation ring of F containing k. Then O admits local uniformization over k.
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Lychrel numbers in base 10
Lychrel conjecture (base 10): conjecturally, there are no Lychrel numbers in base 10. Equivalently, every positive integer eventually becomes a palindrome under the Lychrel iteration.
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Magic Squares
Does there exist a 3 × 3 matrix such that every entry is a distinct square, and all rows, columns, and diagonals add up to the same value? 0 is excluded, as a Magic Square of Squares with 0 and 8 distinct squares is know is knownn. See Magic Square of Squares
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Mahler's 3/2 Problem
The Mahler Conjecture states that there are no Z-numbers.
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Main conjecture on fusible numbers
If x is a fusible number and y is its successor, then the interval [x + 1, y + 1) can be divided into intervals [ℓₙ, ℓₙ₊₁), such that the fusible numbers in [ℓₙ, ℓₙ₊₁) are obtained by fusing the n + 1st successor of x with a fusible number.
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Mathoverflow 17560
If 2^x and 3^x are integers, then x must be an integer.
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Mathoverflow 21003
Is there any polynomial f(x, y) ∈ ℚ[x, y] such that f : ℚ × ℚ → ℚ is a bijection?
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Mathoverflow 235893
Assume for n>1, f:ℝ^n→ℝ^n is a bijection, where ℝ^n is equipped with the standard topology. Does the connectedness of (the induced power set map) f imply that of f^-1?
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Mathoverflow 339137
Let P(x), Q(x) ∈ ℝ[x] be two monic polynomials with non-negative coefficients. If R(x) = P(x)Q(x) is a 0,1 polynomial (coefficients only from 0,1), then P(x) and Q(x) are also 0, 1 polynomials.
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Mathoverflow 34145
Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?
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Mathoverflow 75792
Is 2n the complexity of 2^n for 0 < n?
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Maximum exponent in the prime factorization of n
Are there composite numbers n > 4 such that n ≡ a(n) pmodφ(n)? - Thomas Ordowski, Dec 02 2019 This question is equivalent to Lehmer's totient problem LehmerTotient.lehmer_totient; a positive answer here falsifies the universal statement asked about in Erdos828.erdos_828.variants.lehmer_conjecture.
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Mean value problem
Given a complex polynomial p of degree d ≥ 2 and a complex number z there is a critical point c of p, such that |p(z)-p(c)|/|z-c| ≤ |p'(z)|.
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Minimum modulus for the unique multiset-sum problem
Conjecture 1 (Fonollosa, 2026). For every n ≥ 2 and every N < 2^n - 2^⌊ log_2 n⌋, no set of n residues mod N is valid. Equivalently the super-increasing set 2^k - 1 : 0 ≤ k ≤ n-1 attains the least valid modulus, which is minModulus n.
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Monochromatic quantum graphs (inherited vertex colorings)
For N = 6 and all D ≥ 3, does there exist no solution to the monochromatic quantum graph equation system over ℂ?
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Moser's Worm
Moser's Worm Problem What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?
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Moving Sofa Problem
Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion. The motion is needed: horizontalHallway is (-∞, 1] × [0, 1], so a leftward translate of any moving sofa is again one, obtained by sliding right and then following the original motion.
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Multiplicative order of 2 mod 2n+1
If p is an odd prime then a((p^3-1)/2) = p · a((p^2-1)/2). Because otherwise a((p^3-1)/2) < p · a((p^2-1)/2) iff a((p^3-1)/2) = a((p-1)/2) for a prime p. Equivalently p^3 divides 2^p-1-1, but no such prime p is known. - Thomas Ordowski, Feb 10 2014
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No powers as partition numbers
There are no partition numbers a(k) of the form x^m, with x,m integers >1. See comment by Zhi-Wei Sun (Dec 02 2013).
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Non-Power-of-2 Almost Perfect Numbers Conjecture
Non-Power-of-2 Almost Perfect Numbers Conjecture. Does there exist an almost perfect number that is not a power of 2?
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Normality of Irrational Algebraic Numbers
The strong normality conjecture: every irrational algebraic real is absolutely normal.
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Normality of mathematical constants
π is normal in base 10.
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Number of Abelian cubes of length 3n over an alphabet of size 3
Conjecture (Peter Bala, 2022): The supercongruences a(n · p^k) ≡ a(n · p^k-1) pmodp^3k hold for the integer-indexed extension a(n) for all n ∈ ℤ ∖ 0, primes p ≥ 5, and k ≥ 1.
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Number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes
Conjecture: all items for n ≥ 4 are greater than or equal to 1. This is a stronger conjecture than the Goldbach conjecture.
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Number of different products of subsets of 1, 2, …, n
Conjecture: let p ≤ n be prime. If m and p^a m are two such products, then so is p^k m for all 0 < k < a. - Yan Sheng Ang, Feb 13 2020
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Number of digits of n raised to the power of the sum of the digits of n
n=1 and 32 are two fixed points. Are there any others?
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Number of odd primes between n^2 and (n+1)^2 with (n/p) = 1
Conjecture: a(n) > 0 for all n > 0. - _Zhi-Wei Sun_, Dec 29 2012
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Number of pairs of twin primes between n^2 and (n+1)^2
It is conjectured that a(n)>0 for all n>122. Proving this would also prove Legendre's conjecture that there is a prime between n^2 and (n+1)^2. - _T. D. Noe_, Feb 28 2007
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Number of prime powers q<=n such that also q+2 is a prime power
(25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes, conjecture: there are more (but not < 10^6).
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Number of prime powers strictly between n-th prime and (n+1)-th prime
It is conjectured that a(n) ≤ 2 for all n.
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