Erdős Problem #98
Let h(n) be such that any n points in ℝ^2, with no three on a line and no four on a circle, determine at least h(n) distinct distances. Does h(n)/n→ ∞?
No claims yet Be the first →
Formal Conjectures: open problems with Lean statements
Formal Conjectures is an open repository, started by Google DeepMind, of conjectures stated in Lean 4 with Mathlib. Every open problem from it that we import keeps its exact Lean statement, so a proof submitted here is checked by the Lean kernel against that statement. The collection covers Erdős problems, OEIS conjectures, Ben Green's open problems, Wikipedia's lists of unsolved problems, MathOverflow questions and more.
Source: google-deepmind/formal-conjectures. Licence: Apache License 2.0.
Erdős Problem #982
If n distinct points in ℝ^2 form a convex polygon then some vertex has at least lfloorn/2⌋ different distances to other vertices.
No claims yet Be the first →
Erdős Problem #985
Is it true that, for every prime p, there is a prime q ≤ p which is a primitive root modulo p?
No claims yet Be the first →
Erdős Problem #99
For sufficiently large n, is it the case that any set of n points with minimum distance 1 that minimizes diameter must contain an equilateral triangle of side length 1?
No claims yet Be the first →
Erdős Problem #995
Erdős Problem 995: For every lacunary sequence (n_k) of integers and every f ∈ L^2([0,1]) with ∫_0^1 f = 0, is it true that for almost all α, Σ_k < N f(α n_k) = o (N √(loglog N))?
No claims yet Be the first →
Erdős Problem #996
Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?
No claims yet Be the first →
Euclid Numbers conjecture
It is not known whether there is an infinite number of prime Euclid numbers.
No claims yet Be the first →
Euclid-Mullin sequence
"Does the sequence ... contain every prime? ... [It] was considered by Guy and Nowakowski and later by Shanks, [Wagstaff93] computed the sequence through the 43rd term. The computational problem inherent in continuing the sequence further is the enormous size of the numbers that must be factored.
No claims yet Be the first →
Euler's sum of powers conjecture
Euler's sum of powers conjecture states that for integers n > 1 and k > 1, if the sum of n positive integers each raised to the k-th power equals another integer raised to the k-th power, then n ≥ k. The conjecture is known to be false for k = 4 and k = 5, but remains open for k ≥ 6.
No claims yet Be the first →
Expansion of (1 - x)/(1 - 2 x + 3 x^2)
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017; Arno Berger, email, Jan 06 2017]. - N. J. A. Sloane, Feb 08 2017
No claims yet Be the first →
Exponentials conjectures and theorems
Four exponentials conjecture Let x_0, x_1 and y_0, y_1 be ℚ-linearly independent pairs of complex numbers, then some e^x_i y_j is transcendental.
No claims yet Be the first →
Factorial distance to nearest square
This sequence suggests that the distance between a factorial and the closest power is tightly bounded.
No claims yet Be the first →
Factorial primes
There are infinitely many factorial primes.
No claims yet Be the first →
Feit-Thompson conjecture on primes
There are no distinct primes p and q such that q^p - 1/q - 1 divides p^q - 1/p - 1
No claims yet Be the first →
Fermat-Catalan conjecture
The Fermat–Catalan conjecture states that the equation a^m + b^n = c^k has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (a^m, b^n, c^k) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying frac 1 m + frac 1 n + frac 1 k < 1.
No claims yet Be the first →
Fibonacci Primes
There are infinitely many Fibonacci primes, i.e., Fibonacci numbers that are prime It is also a barrier to defining a benchmark from this paper: https://arxiv.org/html/2505.13938v1 (see Figure 8).
No claims yet Be the first →
Finite generation of the cohomology of finite-dimensional Hopf algebras
Finite generation conjecture (Etingof–Ostrik, Conjecture 2.18, algebra part). For every finite-dimensional Hopf algebra A over a field k, the cohomology ring H^(A, k) = Ext^_A(k, k) is a finitely generated k-algebra.
No claims yet Be the first →
Firoozbakht's conjecture
Firoozbakht's conjecture The inequality sqrt[n+1]p_n+1 < sqrt[n]p_n holds for all prime numbers p_n.
No claims yet Be the first →
First Hardy–Littlewood conjecture
Let P = (m_1, …, m_k) be a tuple of distinct positive even integers. Let π_P(n) denote the number of primes p≤ n such that (p, p + m_1, …, p + m_k) forms an admissible prime constellation.
No claims yet Be the first →
Fortune's Conjecture
Fortune's Conjecture: Every Fortunate number is prime.
No claims yet Be the first →
Four-square conjecture with powers of 2, 3, and 5
Zhi-Wei Sun's Four-Square Conjecture (A308734): Any integer n > 1 can be written as (2^a · 3^b)^2 + (2^c · 5^d)^2 + x^2 + y^2 for nonnegative integers a, b, c, d, x, y.
No claims yet Be the first →
Furstenberg's times p, times q conjectures
Conjecture 1.3 (the × p, × q conjecture): the only atomless Borel probability measure on T which is both T_p- and T_q-invariant is the Lebesgue measure.
No claims yet Be the first →
Gap conjecture
If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least e^sqrt n in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the radius.
No claims yet Be the first →
Gauss circle problem
It is conjectured that the correct bound is |E(r)| = O(r^1/2 + o(1)) [Ha59] Hardy, G. H. (1959). _Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work_(3rd ed.). New York: Chelsea Publishing Company. p. 67 See also https://arxiv.org/abs/2305.03549
No claims yet Be the first →
gcd(numerator(H_n), n!)
Conjecture: Every odd prime occurs as a term in the sequence.
No claims yet Be the first →
Generic and maximal rank of 3-tensors
Friedland's conjecture. In the critical range m_3 ≤ (m_1 - 1)(m_2 - 1), and away from the formats (3, 2p+1, 2p+1), the generic rank of a tensor of format (m_1, m_2, m_3) is the value ⌈ m_1m_2m_3 / (m_1 + m_2 + m_3 - 2) ⌉ predicted by a dimension count [Fri12, Conjecture 5.1].
No claims yet Be the first →
Gilbreath's conjecture
Gilbreath's conjecture Gilbreath's conjecture states that every term in the sequence d^k_0 for k > 0 is equal to 1.
No claims yet Be the first →
Goldbach's conjecture
Can every even integer greater than 2 be written as the sum of two primes?
No claims yet Be the first →
Goodman's conjecture on coefficients of p-valent functions
Goodman's conjecture. For every p-valent normalised function f on the unit disk and every n > p, the n-th coefficient is bounded by the Goodman bound: |b_n| ≤ Σ_k=1^p 2k (n+p)!/(p-k)! (p+k)! (n-p-1)! (n^2-k^2) |b_k|.
No claims yet Be the first →
Gottschalk's surjunctivity conjecture
Gottschalk's surjunctivity conjecture (1973): every group is surjunctive. That is, for every group G and every finite alphabet A, every injective cellular automaton on A^G is surjective.
No claims yet Be the first →
Green's Open Problem 12
Let G be an abelian group of size N, and suppose that A ⊂ G has density α. Are there at least α^15 N^10 tuples (x_1, …, x_5, y_1, …, y_5) ∈ G^10 such that x_i + y_j ∈ A whenever j ∈ i, i+1, i+2? Note: We interpret indices modulo 5.
No claims yet Be the first →
Green's Open Problem 15
Does there exist a Lipschitz function f : ℕ → ℤ whose graph Γ = (n, f(n)) : n ∈ ℕ ⊆ ℤ^2 is free of 3-term progressions?
No claims yet Be the first →
Green's Open Problem 22
If 1, …, N is r-coloured then, for N geqslant N_0(r), there are integers x, y geqslant 3 such that x + y, xy have the same colour. Find reasonable bounds for N_0(r). The goal is to improve upon the Green-Sawhney bound.
No claims yet Be the first →
Green's Open Problem 24
If A is a set of n integers, what is the maximum number of affine translates of the set lbrace 0,1,3 rbrace that A can contain? Conjectured in [Aa19] p.579: (1/3 + o(1)) n^2.
No claims yet Be the first →
Green's Open Problem 25
For which values of k is the following true: whenever we partition [N] = A_1 ∪ … ∪ A_k, |bigcup^k_i=1 (A_i hat+ A_i)| ≥ 1/10 N?
No claims yet Be the first →
Green's Open Problem 27
What is the size of the smallest set A ⊂ ℤ / pℤ (with at least two elements) for which no element in the sumset A + A has a unique representation?
No claims yet Be the first →
Green's Open Problem 28
Suppose that X, Y are two finitely-supported independent random variables taking integer values, and such that X + Y is uniformly distributed on its range. Are X and Y themselves uniformly distributed on their ranges?
No claims yet Be the first →
Green's Open Problem 32
Let p be a prime and let A ⊂ ℤ/pℤ be a set of size ⌊ √(p) ⌋. Is there a dilate of A containing a gap of length 100√(p)?
No claims yet Be the first →
Green's Open Problem 36
Do the following exist, for arbitrarily large n? An abelian group H with |H| = n^2+o(1), together with subsets A_1, ..., A_n, B_1, ..., B_n satisfying |A_i||B_i| ≥ n^2-o(1) and |A_i + B_i| = |A_i||B_i|, such that the sets A_i + B_i are disjoint from the sets A_j + B_k (j ≠ k)?
No claims yet Be the first →
Green's Open Problem 38
Can we improve the best upper bound? The base c must be positive, since =O compares norms.
No claims yet Be the first →
Green's Open Problem 39
If A ⊂ ℤ/pℤ is random, |A| = √(p), can we almost surely cover ℤ/pℤ with 100√(p) translates of A? [Gr24]
No claims yet Be the first →
Green's Open Problem 42
Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
No claims yet Be the first →
Green's Open Problem 44
Sieve [N] by removing half the residue classes mod p_i, for primes 2 leqslant p_1 < p_2 < … < p_1000 < N^9/10. Does the remaining set have size at most 1/10 N? We interpret "half the residue classes" as ⌊ p_i / 2 ⌋.
No claims yet Be the first →
Green's Open Problem 51
Suppose that A ⊂ 𝔽_2^n is a set of density α. What is the largest size of coset guaranteed to be contained in 2A? We phrase this by asking for the exact function F(α, n) giving the maximum dimension of a guaranteed coset.
No claims yet Be the first →
Green's Open Problem 52
Suppose that A ⊂ 𝔽_2^n is a set with an additive complement of size K. Does 2A contain a coset of codimension O_K(1)?
No claims yet Be the first →
Green's Open Problem 53
Suppose that 𝔽_2^n is partitioned in to sets A_1, ..., A_K. Does 2A_i contain a coset of codimension O_K(1) for some i?
No claims yet Be the first →
Green's Open Problem 64
Do there exist infinitely many primes p for which p - 2 has an odd number of prime factors, counted with multiplicity (i.e. Ω(p - 2) is odd)?
No claims yet Be the first →
Green's Open Problem 85
Suppose that A is an open subset of [0, 1]^2 with measure α. Are there four points in A determining an axis-parallel rectangle with area gt c α^2?
No claims yet Be the first →
Green's Open Problem 9
Problem 9 (ii): is r_5(N) ≪ N(log N)^-c?
No claims yet Be the first →
Grimm's conjecture
Grimm's Conjecture If n, n+1, …, n+k-1 are all composite numbers, then there are k distinct primes p_i such that p_i divides n + i for all 0 ≤ i ≤ k-1.
No claims yet Be the first →
Half-Fibonacci sequence
Conjecture (1): The natural density of even terms in the sequence is 1/2.
No claims yet Be the first →
Hall's conjecture
Original Hall's conjecture with exponent 1/2.
No claims yet Be the first →
Hartshorne's conjecture on Vector Bundles
There are no indecomposable vector bundles of rank 2 on ℙ^n for n ≥ 7. This is Conjecture 6.3 in [Har1974].
No claims yet Be the first →
Home primes (OEIS A037274)
Every integer at least two reaches a home prime.
No claims yet Be the first →
Hypothesis H
Schinzel conjecture (H hypothesis) If a finite set of polynomials f_i satisfies both Schinzel and Bunyakovsky conditions, there exist infinitely many natural numbers n such that f_i(n) are primes for all i.
No claims yet Be the first →
Idoneal numbers completeness conjecture
Idoneal numbers completeness conjecture.
No claims yet Be the first →
Infinite Regular Primes
Conjecture: The set of regular primes is infinite.
No claims yet Be the first →
Infinitude of Pell number primes
There are infinitely many prime Pell numbers
No claims yet Be the first →
Infinitude of Wall–Sun–Sun primes
A prime p is a Wall–Sun–Sun prime if and only if L_p ≡ 1 pmodp^2, where L_p is the p-th Lucas number. It is conjectured that there is at least one Wall–Sun–Sun prime.
No claims yet Be the first →
Inscribed square problem
Inscribed square problem Does every Jordan curve admit an inscribed square?
No claims yet Be the first →
Integer part of area of a regular polygon with n sides each of length 1
"Usually (perhaps always?) ⌊ n^2 / (4π) - π / 12 ⌋ for a polygon of circumference n. Note that the area of a circle with circumference C is C^2 / (4π)."
No claims yet Be the first →
Integer part of prime(n)/π(n)
Conjecture: As n → ∞, there are infinitely many n's such that a(n) is greater than a(n+1).
No claims yet Be the first →
Integer values of tan(arctan 1 + arctan 2 + ⋯ + arctan n)
Conjecture (Amdeberhan-Medina-Moll, 2008). For every integer n ≥ 5, the value x_n = tan(arctan 1 + arctan 2 + ⋯ + arctan n) is not an integer.
No claims yet Be the first →
Jacobson Conjecture
The Jacobson conjecture (in its modern form): In a (noncommutative) ring which is left and right Noetherian, the intersection of the powers of the Jacobson ideal is trivial
No claims yet Be the first →
Juggler conjecture
Now form a sequence beginning with any positive integer, where each subsequent term is obtained by applying the operation defined above to the previous term. The Juggler Conjecture states that for any positive integer n, there exists a natural number m such that the m-th term of the sequence is 1.
No claims yet Be the first →
Kaplansky's Conjectures
The zero-divisor conjecture If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors.
No claims yet Be the first →
Köthe conjecture
The Köthe conjecture: In any ring, the sum of two nil left ideals is nil.
No claims yet Be the first →
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.
No claims yet Be the first →
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. -
No claims yet Be the first →
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
No claims yet Be the first →
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.
No claims yet Be the first →
Largest odd divisor of a(n-1) + textrmprime(n)
Is a(33900) the last term equal to 1?
No claims yet Be the first →
Latin Tableau Conjecture
The Latin Tableau Conjecture: If G is the simple graph of a Young diagram, then G is CDS-colorable.
No claims yet Be the first →
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.
No claims yet Be the first →
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?
No claims yet Be the first →
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?
No claims yet Be the first →
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.
No claims yet Be the first →
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
No claims yet Be the first →
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.
No claims yet Be the first →
Legendre's conjecture
Does there always exist at least one prime between consecutive perfect squares?
No claims yet Be the first →
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.
No claims yet Be the first →
Lehmer's totient problem
Does there exist a composite number n > 1 such that Euler’s totient function φ(n) divides n - 1?
No claims yet Be the first →
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".
No claims yet Be the first →
Lemoine's conjectures
For all odd integers n ≥ 7 there are prime numbers p,q such that n = p+2q.
No claims yet Be the first →
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.
No claims yet Be the first →
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.
No claims yet Be the first →
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.
No claims yet Be the first →
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
No claims yet Be the first →
Mahler's 3/2 Problem
The Mahler Conjecture states that there are no Z-numbers.
No claims yet Be the first →
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.
No claims yet Be the first →
Mathoverflow 17560
If 2^x and 3^x are integers, then x must be an integer.
No claims yet Be the first →
Mathoverflow 21003
Is there any polynomial f(x, y) ∈ ℚ[x, y] such that f : ℚ × ℚ → ℚ is a bijection?
No claims yet Be the first →
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?
No claims yet Be the first →
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.
No claims yet Be the first →
Mathoverflow 34145
Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?
No claims yet Be the first →
Mathoverflow 75792
Is 2n the complexity of 2^n for 0 < n?
No claims yet Be the first →
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.
No claims yet Be the first →
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)|.
No claims yet Be the first →
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.
No claims yet Be the first →
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?
No claims yet Be the first →