Sum of Fermat number and Mersenne number minus 1: 2^2^n + 2^n - 1
The first 5 entries are primes. Are there infinitely many primes in this sequence?
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.
Sum of four squares with square conditions
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.
No claims yet Be the first →
Sum of next n primes
The only positive integer n such that a(n) is a perfect square is n=38. - Carlos Eduardo Olivieri, Mar 09 2015
No claims yet Be the first →
Sum of squares of divisors of n
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
No claims yet Be the first →
Sum of squares of nonacci numbers
Primes in this sequence include: a(8) = 2, which is next?
No claims yet Be the first →
Sum of three cubes
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.
No claims yet Be the first →
Sum of two numbers with prime conditions
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.
No claims yet Be the first →
Sum of two squares, a power of 3, and a power of 5
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.
No claims yet Be the first →
Tarski's exponential function problem
Tarski's exponential function problem. Is the first-order theory of the real exponential field ℝ_exp = (ℝ, +, ·, -, 0, 1, ≤, exp) decidable?
No claims yet Be the first →
Taxicab numbers
Taxicab number for k=5, m=2, and n=2 is not known. Whether such a number exists is also not known.
No claims yet Be the first →
The 1680-Conjecture
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.
No claims yet Be the first →
The Alon-Tarsi short cycle cover conjecture
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)|.
No claims yet Be the first →
The Andrews-Curtis conjecture
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.
No claims yet Be the first →
The Auslander-Reiten conjecture
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.
No claims yet Be the first →
The Bing-Borsuk Conjecture
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).
No claims yet Be the first →
The Catch-Up game and conjecture
Let T_N = Σ_k=1^N k = N(N+1)/2. If T_N is even (equivalently N ≡ 0 pmod 4 or N ≡ 3 pmod 4), then under optimal play the game Catch-Up(1, …, N) ends in a draw.
No claims yet Be the first →
The circulant Hadamard conjecture
Every circulant Hadamard matrix has order at most four.
No claims yet Be the first →
The Curling Number Conjecture
The sequence will eventually reach 1.
No claims yet Be the first →
The Eisenbud-Green-Harris conjecture
The Eisenbud-Green-Harris conjecture. Let I ⊆ k[x_1, …, x_n] be a homogeneous ideal containing a regular sequence of forms of degrees d_1 ≤ … ≤ d_c. Then there is a lex ideal L such that I has the same Hilbert function as L + (x_1^d_1, …, x_c^d_c).
No claims yet Be the first →
The Erdős–Moser equation
The only positive solution of S_k(m)=m^k is (k,m)=(1,3).
No claims yet Be the first →
The first Atiyah–Sutcliffe conjecture
Atiyah–Sutcliffe Conjecture 1, stated as Conjecture 1.1 in Mazur–Petrenko: the configuration polynomials are linearly independent.
No claims yet Be the first →
The Gerstenhaber problem for three commuting matrices
The Gerstenhaber problem: if A, B, and C are pairwise commuting n × n matrices over a field K, is the dimension of the unital K-algebra K[A, B, C] they generate always at most n?
No claims yet Be the first →
The Goormaghtigh conjecture
The only Goormaghtigh numbers are 31 and 8191.
No claims yet Be the first →
The Hasse–Weil conjecture for elliptic curves
The L-series of an elliptic curve over a number field has a meromorphic continuation to ℂ.
No claims yet Be the first →
The length of an s-increasing sequence of r-tuples
F(n) ≤ n^3/2.
No claims yet Be the first →
The Lovász–Plummer conjecture (proved 2011) and Sheehan's conjecture
Sheehan's conjecture (1977). Every 4-regular graph with a Hamiltonian cycle has a second Hamiltonian cycle (one with a different edge set).
No claims yet Be the first →
The prime numbers
Conjecture from Thomas Ordowski (2023): log log a(n+1) - log log a(n) < 1/n for n > 0.
No claims yet Be the first →
The Rule 30 Prize Problems
Rule 30 Prize, Problem 1 (non-periodicity). The center column of Rule 30 is not eventually periodic: there is no positive period p and threshold N past which the column repeats with period p.
No claims yet Be the first →
The S_3-conjecture (conjugacy classes of distinct sizes)
Markel's S_3-conjecture (1973): any nontrivial finite ah-group is isomorphic to S_3. The conjecture is open in general; it is known to be true for solvable groups.
No claims yet Be the first →
The small Cohen-Macaulay modules conjecture
The small Cohen-Macaulay modules conjecture. If R is a complete Noetherian local ring, then there is a finitely generated R-module M ≠ 0 such that some system of parameters of R is a regular sequence on M. Hochster stated the conjecture for complete local domains [Ho17, Conjecture 2.1].
No claims yet Be the first →
The symbol length of K^M_n(ℂ(x_1, …, x_m))/p
The symbol length problem for complex rational function fields [Krashen2024, Problem 2.1.3.12 and §2.1.3.4]: determine, as a function of m, n and the prime p, the symbol length of K^M_n(ℂ(x_1, …, x_m))/p, that is the least k such that every class is a sum of at most k symbols, or ∞ if there is no…
No claims yet Be the first →
Toronto spaces
Any T2, Toronto space is discrete.
No claims yet Be the first →
Trajectory of 103 under the Reverse and Add! operation in base 3
103 is conjectured to be the smallest number such that the Reverse and Add! algorithm in base 3 does not lead to a palindrome. Its trajectory is conjectured to never reach a palindrome.
No claims yet Be the first →
Triangular matchstick numbers in the class of prime numbers
Terms are squares at only(?) three values of n = 3, 6, 4072: corresponding terms are 6^2, 13^2, and 15735^2.
No claims yet Be the first →
Tug of war score between prime gap increases and decreases
Is the score a(n) > 0 for some n > 250000?
No claims yet Be the first →
Twin prime conjecture
Are there infinitely many primes p such that p + 2 is prime?
No claims yet Be the first →
TxGraffiti Conjecture 2: zero forcing versus independence (cubic graphs)
TxGraffiti Conjecture 2: for every connected graph G with Δ(G) ≤ 3 and G ≠ K_4, Z(G) ≤ α(G) + 1. This conjecture is open.
No claims yet Be the first →
TxGraffiti Conjecture 3: independent domination versus saturation (regular)
TxGraffiti Conjecture 3: for every r-regular graph G (r ≥ 1), i(G) ≤ μ^(G). This conjecture is open**.
No claims yet Be the first →
Unique Crystal Components
If n = ab is a crystal, then there are no other pairs of positive integers c, d > 1, different from the couple a, b, such that n = cd and B(c, d) ∈ ℕ, i.e., the components of the crystals are unique.
No claims yet Be the first →
Value of n-th cyclotomic polynomial at n
a(28341) is divisible by 283411^2. What is the next n such that a(n) is not squarefree?
No claims yet Be the first →
Vaught conjecture
The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite, aleph_0 or 2^aleph_0.
No claims yet Be the first →
VCₙ dimension of convex sets in ℝⁿ, ℝⁿ⁺¹, ℝⁿ⁺²
Every convex set in ℝ^3 has VC_2 dimension at most 2.
No claims yet Be the first →
Vizing's conjecture (1968)
Vizing's conjecture (1968). For all finite simple graphs G and H, the domination number of the Cartesian (box) product satisfies γ(G square H) ≥ γ(G) γ(H).
No claims yet Be the first →
Weak tiling problems
Problem 4.1. Let Ω ⊂ ℝ be a finite union of intervals and ν a weak tiling measure for Ω. Must supp(ν) have bounded density?
No claims yet Be the first →
Wieferich primes
There are infinitely many Wieferich primes.
No claims yet Be the first →
Wilson primes
There are infinitely many Wilson primes.
No claims yet Be the first →
Wolstenholme numbers
Conjecture: for n > 3, gcd(n, a(n-1)) = A089026(n). - Amiram Eldar and Thomas Ordowski, Jul 28 2019
No claims yet Be the first →
Wolstenholme Prime
It is conjectured that there are infinitely many Wolstenholme primes. Reference: Wikipedia
No claims yet Be the first →
Woodall Primes
There are infinitely many prime numbers of the form k * 2 ^ k - 1 for k > 1.
No claims yet Be the first →
Written on the Wall II - Conjecture 100
WOWII Conjecture 100 (status O): For a simple connected graph G, α(G) ≤ ⌈(max_v l(v) + 0.5 · degreeL2Norm(Gᶜ)) / 2⌉ where α(G) = G.indepNum is the independence number, max_v l(v) is the maximum over all vertices of the independence number of the neighbourhood (in G), and degreeL2Norm(Gᶜ) is the…
No claims yet Be the first →
Written on the Wall II - Conjecture 133
WOWII Conjecture 133: For a simple connected graph G, path(G) ≥ rad(G) + (avg_v l(v))^cC_4(G), where path(G) is the path number of the graph (number of vertices of a largest induced path), rad(G) is the radius (minimum eccentricity, as a natural number), avg_v l(v) = l(G) is the average…
No claims yet Be the first →
Written on the Wall II - Conjecture 19
WOWII Conjecture 19 If G is connected then the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ FLOOR((∑ ecc(v))/(|V|) + sSup (range (l G))), where ecc(v) denotes eccentricity and l(G) is the independence number of neighbourhoods.
No claims yet Be the first →
Written on the Wall II - Conjecture 198a
WOWII Conjecture 198a For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path. Here b(G) is the number of vertices in a largest induced bipartite subgraph, and ecc_avg(G) is the average eccentricity of G.
No claims yet Be the first →
Written on the Wall II - Conjecture 40
WOWII Conjecture 40 For a nontrivial connected graph G the size f(G) of a largest induced forest satisfies f(G) ≥ ceil((p(G) + b(G) + 1)/2) where p(G) is the path cover number and b(G) is the largest induced bipartite subgraph size.
No claims yet Be the first →
Written on the Wall II - Conjecture 61
WOWII Conjecture 61 For a simple connected graph G, the size f(G) of a largest induced forest satisfies f(G) ≥ residue(G) + ⌈ diam(G) / 3 ⌉, where residue(G) is the Havel-Hakimi residue and diam(G) is the diameter of G.
No claims yet Be the first →
Zagier's Conjecture on Multiple Zeta Values
Zagier's conjecture The ℚ-dimension of the vector space spanned by all multiple zeta values of weight n equals d_n, where d_n is the Zagier dimension sequence satisfying d_0 = 1, d_1 = 0, d_2 = 1, and d_n = d_n-2 + d_n-3 for n ≥ 3.
No claims yet Be the first →
Zariski Cancellation
The Zariski Cancellation Problem: every polynomial ring over a field k of characteristic 0 is cancellative.
No claims yet Be the first →