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).
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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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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.
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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.
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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.
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Fortune's Conjecture
Fortune's Conjecture: Every Fortunate number is prime.
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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.
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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.
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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.
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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
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gcd(numerator(H_n), n!)
Conjecture: Every odd prime occurs as a term in the sequence.
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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].
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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.
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Goldbach's conjecture
Can every even integer greater than 2 be written as the sum of two primes?
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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|.
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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.
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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.
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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?
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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.
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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.
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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?
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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?
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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?
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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)?
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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)?
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Green's Open Problem 38
Can we improve the best upper bound? The base c must be positive, since =O compares norms.
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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]
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Green's Open Problem 42
Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
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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 ⌋.
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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.
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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)?
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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?
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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)?
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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?
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Green's Open Problem 9
Problem 9 (ii): is r_5(N) ≪ N(log N)^-c?
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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.
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Half-Fibonacci sequence
Conjecture (1): The natural density of even terms in the sequence is 1/2.
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Hall's conjecture
Original Hall's conjecture with exponent 1/2.
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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].
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Home primes (OEIS A037274)
Every integer at least two reaches a home prime.
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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.
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Idoneal numbers completeness conjecture
Idoneal numbers completeness conjecture.
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Infinite Regular Primes
Conjecture: The set of regular primes is infinite.
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Infinitude of Pell number primes
There are infinitely many prime Pell numbers
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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.
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Inscribed square problem
Inscribed square problem Does every Jordan curve admit an inscribed square?
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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)."
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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).
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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.
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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
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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.
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