Bloch and Landau constants
Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.
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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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Bondy's conjecture on longest cycles in highly connected graphs
Conjecture 1 (Bondy, 1980). Let k ≥ 1 and let G be a k-connected graph on n vertices. If δ(G) ≥ n + k(k-1)/k+1, then for every longest cycle C of G, every path in G - V(C) has at most k-1 vertices.
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Borsuk's conjecture
Borsuk's conjecture, open range: every bounded subset of ℝ^n with at least two points can be partitioned into n + 1 sets of strictly smaller diameter, for 4 ≤ n ≤ 62. The conjecture is known to be true for n ≤ 3 and false for n ≥ 63.
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Brennan's Conjecture
Brennan's conjecture, part 1: B(-2) = 1.
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Brocard's Conjecture
Brocard's Conjecture For every n ≥ 2, between the squares of the n-th and (n+1)-th primes, there are at least four prime numbers.
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Büchi's problem
Büchi's problem There exists a positive integer M such that, for all integers x and a, if (x+n)^2 + a is a square for M consecutive values of n, then a = 0.
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Bugeaud Collection of Conjectures and Open Questions: Confined Powers of Non-Pisot Numbers
Problem 10.7. Let ε be a positive real number. Are there arbitrarily large real numbers α such that α is not a Pisot number and all the fractional parts α^n, n ≥ 1, are lying in an interval of length ε / α? [Bug12b]
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Bugeaud Collection of Conjectures and Open Questions: Fractional Parts of Powers
Problem 10.1. Are there a transcendental number α and a positive real number ξ such that lVert ξ α^n rVert tends to~0 as~n tends to infinity? [Har19] (Trivial for |α| < 1)
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Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers
Problem 10.9. There are no real numbers ξ such that 0 ≤ ξ (3/2)^n < 1/2 for every positive integer n, i.e. no Z-number exists. Posed by Mahler [Mah68].
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Bugeaud Collection of Conjectures and Open Questions: p-adic Littlewood Conjecture
Problem 10.8 (p-adic Littlewood conjecture). For every real number ξ and every prime number p, inf_q ≥ 1 q · lVert q ξ rVert · |q|_p = 0, where lVert · rVert denotes the distance to the nearest integer and |·|_p denotes the p-adic absolute value. Posed by de Mathan and Teulié [dMT04].
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Bugeaud Collection of Conjectures and Open Questions: Pisot orbits on the Cantor set
Problem 10.61. Let α > 2 be a Pisot number. For every ξ ∈ C(α) the sequence (ξ α^n)_n ≥ 1 is not uniformly distributed modulo one.
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Bunyakovsky conjecture
Bunyakovsky conjecture If a polynomial f over integers satisfies both Schinzel and Bunyakovsky conditions, there exist infinitely many natural numbers m such that f(m) is prime.
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Busy Beaver
Determine the value of the Busy Beaver function at n = 6.
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Can a prime p satisfy 2^p-1 ≡ 1 pmodp^2 and 3^p-1 ≡ 1 pmodp^2?
Can a prime p satisfy 2^p-1 ≡ 1 pmodp^2 and 3^p-1 ≡ 1 pmodp^2 simultaneously? That is, does there exist a prime p that is both a Wieferich prime and a Mirimanoff prime? Wikipedia's list of unsolved problems poses this question, citing J. B. Dobson, On Lerch's formula for the Fermat quotient.
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Carmichael's totient function conjecture
Carmichael's totient function conjecture: For every positive natural number n, there exists a natural number m with m ≠ n, such that φ(n) = φ(m).
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Casas-Alvero Conjecture
The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.
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Catalan-Mersenne numbers
Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime.
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Catalan's conjecture and related Diophantine equations
For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation ax^n - by^m = c where (m, n) ≠ (2, 2) and x, y > 1.
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Central binomial sum a(n) = Σ_k=0^n (-4)^k C(n, k)^2 C(n-k, k)^2
If p is a prime with p ≡ 1, 9 pmod20 and p = x^2 + 5y^2 with x, y integers, then Σ_k=0^p-1 a(k) ≡ 4x^2 - 2p pmodp^2. - _Zhi-Wei Sun_, Jul 01 2010
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Central binomial sum a(n) = Σ_k=0^n C(n, k)^2 C(n-k, k)^2 (-16)^k
If p is a prime with (p/7) = 1 and p = x^2 + 7y^2 with x, y integers, then Σ_k=0^p-1 (-1)^k a(k) ≡ 4x^2 - 2p pmodp^2. - _Zhi-Wei Sun_, Jul 17 2010
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Central trinomial coefficients
An integer n > 3 is prime if and only if a(n) ≡ 1 pmodn^2. We have verified this for n up to 8 · 10^5, and proved that a(p) ≡ 1 pmodp^2 for any prime p > 3 (cf. A277640). - Zhi-Wei Sun, Nov 30 2016
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Chua's Euclidean prime sequence
Does Chua's sequence contain every prime?
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Chvátal's Conjecture
If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.
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Class number problem for real quadratic fields
There are infinitely many real quadratic fields ℚ(√d) with class number one, where d > 1 is a squarefree integer.
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Coefficients of Π_k>0 (1 - x^k/k!)
The coefficients c(n) of A(x)^2 = (Σ_n ≥ 0 a(n) x^n)^2 differ in sign from c(n-1) if and only if n is a triangular number. - _Peter Bala_, Mar 17 2022
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Collatz step differences
Conjecture 1: More than half of the terms are 0. - _Ya-Ping Lu_, May 04 2024
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Concatenation of the next n numbers
"The second term is a prime. When is the next prime, if there is another? - _N. J. A. Sloane_, Dec 16 2016"
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Congruent Number
Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers.
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Conjecture 1.35(c)
Do there exist simple pro-orderable groups?
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Conjecture 1.40
Is a group a nilgroup if it is the product of two normal nilsubgroups? Since H and K are normal, the product HK coincides with the join H sqcup K, so "G is the product of H and K" is stated as H sqcup K = G.
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Conjecture 1.74 (Minimal topological groups)
Describe all minimal topological groups, that is, all non-discrete Hausdorff topological groups whose proper closed subgroups are all discrete.
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Conjecture 19.25
Let G and H be finite groups of the same order with Σ_g ∈ G φ(|g|) = Σ_h ∈ H φ(|h|), where φ is the Euler totient function. Suppose that G is simple. Is H necessarily simple?
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Conjecture 20.76
Let G be a finite p-group and assume that all abelian normal subgroups of G have order at most p^k. Is it true that every abelian subgroup of G has order at most p^2k?
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Conjecture 8.8
Does there exist a non-cyclic finitely presented group G which contains an element a such that each element of G is conjugate to some power of a? Here a power of a means a^n for some n ∈ ℤ.
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Conjecture about cardinality of Lindelöf spaces
Is there a Lindelöf Tychonoff space with singletons as Gδ sets with cardinality greater than the continuum? Note: the cited paper uses a blanket convention that all spaces are Tychonoff.
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Conjectures about Latin Squares
Conjecture 3.2 in [Wa2011]: Each Latin square of odd order has at least one transversal.
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Conjectures about Mersenne primes
For any odd natural number p if two of the following conditions hold, then all three must hold: 1. 2^p-1 is prime 2. (2^p+1)/3 is prime 3. Exists a number k such that p = 2^k pm 1 or p = 4^k pm 3
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Conjectures about the Mandelbrot and Multibrot sets
The MLC conjecture, stating that the mandelbrot set is locally connected.
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Conjectures about Weakly First Countable spaces
Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space X such that 𝔠 < |X|. Note: [Ar2013] uses a blanket convention that all spaces are Tychonoff and "compact" means compact Hausdorff.
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Conjectures around homogeneous topological spaces
Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?
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Conjectures associated with A038552
All terms of A038552 are congruent to 19 pmod24.
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Conjectures associated with A063880
All members of the sequence satisfy n ≡ 108 pmod216.
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Conjectures associated with A067720
For members of the sequence other than 8, we have k + 1 is prime.
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Conjectures associated with A100474
After a(2) = 5, is there another prime?
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Conjectures associated with A100800
A100800 Conjecture: No term is zero.
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Conjectures associated with A101779
It is conjectured k always exists.
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Conjectures associated with A105210
Cormier and Selfridge found 5 starting values for which the sequences appear to not merge. The sequences were checked up to 10^8.
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Conjectures associated with A109227
Conjecture: a(2) and a(121) are primes. Are there any more?
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Conjectures associated with A109671
Does the sequence contain every positive integer (cf. A169741)?
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Conjectures associated with A109845
Conjecture: There are infinitely many primes in this sequence.
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