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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For a graph G, we define Δ(G) to be the maximum degree, ω(G) to be the size of the largest clique subgraph, and χ(G) to be the chromatic number. Reed's omega, delta, and chi conjecture states that χ(G) ≤ ⌈ 1/2(ω(G) + Δ(G) + 1) ⌉.
On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n) = 0 if and only if n < 16 or n ∈ 18, 21, 24, 51, 84, 1011, 59586; in other words, except for 35, 41, 47, 101, 167, 2021, 119171, any odd integer greater than 30 can be written as the sum of a prime congruent to 1 bmod 6, a positive power of 2 and…
Zhi-Wei Sun's Conjecture (A232174): Any integer n > 1 can be written as x + y with x, y > 0 such that both x + ny and x^2 + ny^2 are prime.
Resolution of singularities in positive characteristic. Let k be a perfect field of characteristic p > 0 and let X be an integral scheme that is separated and of finite type over k. Then there is an integral scheme Y that is smooth over k together with a proper birational morphism Y → X.
It is conjectured that the integer k = 509203 is the smallest Riesel number, that is, the first n such that a(n) = -1 is 254602.
For any tree T with n edges, the complete graph K_2n+1 decomposes into 2n+1 edge-disjoint copies of T. A "copy" of T is the image T.map(f_i) of T under a vertex embedding f_i : V hookrightarrow Fin(2n+1); the copies are pairwise edge-disjoint and together cover every edge of K_2n+1.
Conjecture: Every record of differences a(n)-a(n-1) more than 5 is the greater of twin primes (A006512).
Rudin's conjecture. The maximal number of squares among the first N terms of a non-trivial arithmetic progression grows at most like √(N): Q(N) = O(√(N)).
Given any set of n complex numbers z_1, ..., z_n that are linearly independent over ℚ, the field extension ℚ(z_1, ..., z_n, e^z_1, ..., e^z_n) has transcendence degree at least n over ℚ.
The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that for every positive integer n, the addition-chain length of 2^n - 1 is at most n - 1 + ℓ(n).
PSW conjecture (Selfridge's test) Let p be an odd number, with p ≡ ± 2 pmod5, 2^p-1 ≡ 1 pmodp and F_p+1 ≡ 0 pmodp, then p is a prime number.
Positivity conjecture. Let R be a regular local ring and let M, N be finitely generated R-modules such that M otimes_R N has finite length. If dim M + dim N = dim R, then χ(M, N) > 0. The hypothesis on dimensions forces M and N to be nonzero, since the dimension of the zero module is bot.
Serre's uniformity question over ℚ [Ser72, Lem17]: is there a bound C such that every non-CM elliptic curve over ℚ has surjective mod-p Galois representation for every prime p > C?
Sidorenko's conjecture (1993). For every finite bipartite simple graph H and every finite simple graph G: t(H, G) ≥ t(K_2, G)^e(H), where K_2 denotes the single-edge graph on 2 vertices (i.e. completeGraph (Fin 2)).
The Sierpiński problem (Selfridge's conjecture). Is 78557 the smallest Sierpiński number? Selfridge conjectured that 78557 is the smallest Sierpiński number.
Singmaster's conjecture: the number of times any number t > 1 appears in Pascal's triangle is bounded.
Conjecture: liminf_n → ∞ a(n)/p_n+1^2 = 1 < limsup_n → ∞ a(n)/p_n+1^2 = 2. - Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015
It is conjectured that a(24) = 0 since no factorial less than 10000 contained just 24 sixes.
Conjecture: the sequence is infinite, that is, for every n ≥ 1 there is some k > n with S(n) | S(k), so that a(n) is defined.
There is a conjecture that the first zero is n = 65536 = 2^16 (which is equivalent to the statement that 2^2^k + 1 is composite for k > 4). - _T. D. Noe_, Feb 25 2011
Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n.
Conjecture 2: For any k ≥ 3, there are infinitely many primes of the form n^k + m^k + 1 for n, m ≥ 1. - _Ulrich Krug_, 2009
Conjecture: for every n > 1 there exists a number k < n such that nk + 1 is a prime.
Conjecture: a(n) = O(n^3). The source defines a(n) as the least m with 2^n - m and 2^n + m prime, so it implicitly asserts that such an m exists. Since a n = 0 when no such m exists, the existence of a prime pair is stated explicitly for all sufficiently large n.
There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number.
"Conjecture: a(n) < n^2 for n > 1. - _Thomas Ordowski_, Dec 19 2016"
Conjecture: There are infinitely many composite numbers n such that a(n) is nonzero.
What is the smallest integer m > 1 such that a(10^m) is nonzero? - _Farideh Firoozbakht_, Jan 07 2015
At present, the 0 entry for n = 5 is only a conjecture. That is, it is conjectured that there is no positive integer x such that σ_1(x) bmod x = 5.
Is 10 a solitary number? The smallest positive integer whose solitary status is currently unresolved is 10, with abundancy index σ(10) / 10 = 9/5.
Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their heights, then 50% of the curves have rank 0 and 50% have rank 1. See p. 28 of https://people.maths.bris.ac.uk/~matyd/BSD2011/bsd2011-Bhargava.pdf.
Wichmann's conjecture on optimal rulers. Every optimal ruler of sufficiently large length is a Wichmann ruler W(r, s) (up to reflection, i.e. reversing the segment list). Posed by Wichmann [Wi63].
[KLM2023, Problem 7.1] asks whether a bounded, measurable, nowhere dense subset Ω ⊂ ℝ^d of positive measure can be spectral. The answer is known to be negative for d = 1, so the dimension is restricted to d ≥ 2, where the problem is open.
Construct an S(t, k, n)-Steiner system with n > k > t > 5, t < 10, and n < 200. No example of a Steiner system with t > 5 is known, despite a 2014 existence theorem by Keevash showing that such systems must exist for sufficiently large n. Reference: Large Steiner Systems
Strong Sensitivity Conjecture, for every Boolean function f : 0,1^n → 0,1, bs(f) ≤ s(f)^2. We call this the strong sensitivity conjecture because the original sensitivity conjecture only asked for a polynomial bound in terms of s(f).
The first 5 entries are primes. Are there infinitely many primes in this sequence?
Zhi-Wei Sun's Conjecture (A281976): Any integer n ≥ 0 can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers and z ≤ w, such that both x and x + 24y are squares.
The only positive integer n such that a(n) is a perfect square is n=38. - Carlos Eduardo Olivieri, Mar 09 2015
Conjecture: For each k = 2,3,..., all the rational numbers σ_k(n)/n^k = Σ_d|n 1/d^k (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015
Primes in this sequence include: a(8) = 2, which is next?
An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if n is not 4 or 5 mod 9.
The conjecture for sequence A231201: for any n > 1, there exist x, y > 0 such that n = x + y and 2^x + y is prime.
Zhi-Wei Sun's Conjecture (A303656): Any integer n > 1 can be written as the sum of two squares, a power of 3, and a power of 5.
Tarski's exponential function problem. Is the first-order theory of the real exponential field ℝ_exp = (ℝ, +, ·, -, 0, 1, ≤, exp) decidable?
Taxicab number for k=5, m=2, and n=2 is not known. Whether such a number exists is also not known.
Zhi-Wei Sun's 1680-Conjecture (A280831): Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x^4 + 1680 y^3 z is a square.
Conjecture 4 (Alon-Tarsi, 1985). Every bridgeless graph has a list of cycles covering every edge, with Σ_C |E(C)| ≤ 7/5|E(G)|.
The Andrews-Curtis conjecture. Every normally generating n-tuple in the free group of rank n is Andrews-Curtis equivalent to the standard tuple of free generators.
The Auslander-Reiten conjecture [AR75]. Let Λ be an Artin algebra and M a finitely generated Λ-module with Ext^i_Λ(M, Λ) = 0 and Ext^i_Λ(M, M) = 0 for all i > 0. Then M is projective.
The Bing-Borsuk Conjecture: every n-dimensional homogeneous absolute neighborhood retract is a topological n-manifold. A topological space X is an n-dimensional manifold when T2Space X ∧ Nonempty (ChartedSpace (Fin n → ℝ) X).
