Remainder of 2^n+2 modulo n
"Conjecture: every integer k ≥ 0 appears in a(n) at least once."
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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.
Representations as p + 2^x + 11 · 2^y with p ≡ 1 pmod 6
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…
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Representations n = k^2 + m (0 < k^2 ≤ m) with σ(k^2) + φ(m) prime
Conjecture (i): a(n) > 0 for all n > 1. - _Zhi-Wei Sun_
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Representations of n = p + q + (n bmod 2)q with Jacobi symbol constraints
Conjecture: a(n) > 0 for all n > 5. - _Zhi-Wei Sun_, Dec 30 2012
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Representations of n as a sum of generalized heptagonal, octagonal, and nonagonal numbers
Conjecture: a(n) > 0 for all n. - _Zhi-Wei Sun_
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Representations of n as the sum of two pentagonal and a generalized decagonal number
Conjecture: a(n) > 0 for all n. - _Zhi-Wei Sun_
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Representations of n as x^2 + φ(y^2) + φ(z^2)
Conjecture (i): a(n) > 0 for all n > 6.
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Representations of n using prime pairs with prime gaps of 6
Conjecture: a(n) > 0 for all even n > 8012 and odd n > 15727.
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Representations using Sophie Germain primes and twin prime pairs
Conjecture (i), first part: a(n) > 0 for all n > 1.
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Representations with prime conditions
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.
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Resolution of singularities
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.
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Riesel Problem
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.
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Ringel's Conjecture
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.
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Rowland-style prime-generating recurrence
Conjecture: Every record of differences a(n)-a(n-1) more than 5 is the greater of twin primes (A006512).
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Rudin's conjecture on squares in arithmetic progressions
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)).
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Schanuel's Conjecture
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.
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Scholz conjecture on addition chains
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).
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Selfridge's conjectures
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.
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Sequence designed to show there are infinitely many primes congruent to 1 modulo 4
Similarly to Sylvester's sequence (A000058), it is unknown if all terms are squarefree.
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Serre's multiplicity conjectures
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.
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Serre's uniformity conjecture over the rationals
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?
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Sidorenko's conjecture (1993)
Tournament Anti-Sidorenko (TAS) Trees Conjecture. For every finite undirected tree T, there exists an orientation vecT of its edges such that for any finite tournament G, the homomorphism density satisfies: t_vecT(G) ≤ 2^-e(T) where e(T) is the total number of edges in T.
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Sierpiński number
The Sierpiński problem (Selfridge's conjecture). Is 78557 the smallest Sierpiński number? Selfridge conjectured that 78557 is the smallest Sierpiński number.
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Singmaster's conjecture
Singmaster's conjecture: the number of times any number t > 1 appears in Pascal's triangle is bounded.
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Smallest composite c such that textrmprimorial(n) + c is prime
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
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Smallest deficient multiplier for non-deficiency
Conjecture: For every n ≥ 2, a(n) is 1, 3, or a power of 2.
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Smallest factorial containing exactly n 6's
It is conjectured that a(24) = 0 since no factorial less than 10000 contained just 24 sixes.
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Smallest index k > n such that (p_k+p_k+1)/(p_n+p_n+1) is an integer ≥ 2
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.
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Smallest integer s > n such that 1 + 2s + 3s^2 + … + n s^n-1 is prime
Conjecture: For each n ≥ 2, there are infinitely many primes of the form 1 + 2s + … + n s^n-1 where s is a positive integer.
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Smallest k ≥ 1 such that the concatenation of k threes and prime(n) is prime
Conjecture: For n ≥ 4, a(n) > 0.
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Smallest m ∈ [1, n] such that C(n, m) is evil
Conjecture: there are only five n (0, 1, 2, 7, 8) for which all entries of the n-th Pascal row (A007318) are odious (A000069), which is equivalent to a(n) = 0.
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Smallest m > 0 such that n · 2^m + 1 is prime
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
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Smallest m > 0 such that there are no primes between nm and n(m+1) inclusive.
Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n.
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Smallest m such that n^3 + m^3 + 1 is prime
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
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Smallest n-th power equal to a sum of consecutive immediately preceding positive n-th powers
Is a(n) ≠ 0 for any n > 3?
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Smallest number k such that kn + 1 is prime
Conjecture: for every n > 1 there exists a number k < n such that nk + 1 is a prime.
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Smallest number m such that 2^n - m and 2^n + m are primes
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.
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Smallest palindrome with exactly n divisors
There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number.
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Smallest prime ≡ 1 pmod n
"Conjecture: a(n) < n^2 for n > 1. - _Thomas Ordowski_, Dec 19 2016"
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Smallest prime formed by concatenation n, n-1, …, n-k
Conjecture: There are infinitely many composite numbers n such that a(n) is nonzero.
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Smallest r such that (concatenation of n, r times) · 10 + 1 is prime
What is the smallest integer m > 1 such that a(10^m) is nonzero? - _Farideh Firoozbakht_, Jan 07 2015
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Smallest x such that x bmod φ(x) = n, or 0 if no such x exists
Conjecture: a(n) > 0 for all n.
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Smallest x such that σ(x) bmod x = n
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.
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Smarandache–Wellin primes
Smarandache Problems states: "There are infinitely many primes in the smarandache concatenated prime sequence."
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Solitary Numbers
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.
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Some conjectures about ranks of elliptic curves over ℚ
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.
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Sparse Ruler
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].
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Spectral sets and weak tiling
[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.
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Square root of the absolute value of the determinant of the prime sum indicator matrix
We conjecture that a(n) > 0 for all n > 15.
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Steiner Systems
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
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Strong Sensitivity Conjecture (bs(f) ≤ s(f)^2)
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).
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Sum of d · τ(d) over proper divisors d of n
Conjecture: 21 is the only positive integer n such that a(n) = n.
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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?
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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.
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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
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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
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Sum of squares of nonacci numbers
Primes in this sequence include: a(8) = 2, which is next?
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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.
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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.
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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.
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Sumner's universal tournament conjecture (1971)
Sumner's universal tournament conjecture (1971). Every tournament on 2n - 2 vertices contains every oriented tree on n ≥ 2 vertices as a subgraph.
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Sums of a cube, a square, and a positive triangular number
Conjecture: a(n) > 0 for all n > 0.
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Sums of a square and three weighted cubes
Conjecture (i): For any m ∈ 3, 4, 5, 6 and n ≥ 0, there are nonnegative integers w, x, y, z such that n = w^2 + x^3 + 2y^3 + mz^3.
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Sums of a triangular number and three weighted cubes
Conjecture (i): Any positive integer can be written as w(w+1)/2 + x^3 + by^3 + cz^3 with w > 0 and x, y, z ≥ 0, provided that (b, c) ∈ (1,2), (1,3), (1,4), (1,6), (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (2,20), (2,21), (2,34), (3,3), (3,4), (3,5), (3,6), (4,10).
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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?
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Taxicab numbers
Taxicab number for k=5, m=2, and n=2 is not known. Whether such a number exists is also not known.
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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.
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The Alon-Tarsi short cycle cover conjecture
Conjecture 4 (Alon-Tarsi, 1985), simple-graph specialization. Every bridgeless simple graph has a list of cycles covering every edge, with Σ_C |E(C)| ≤ 7/5|E(G)|.
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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.
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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.
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The Balister–Győri–Schelp conjecture
The Balister–Győri–Schelp conjecture (Conjecture 1): for s ≥ 2, every zero-sum list of 2^s-1 nonzero vectors in 𝔽_2^s is the prescribed-difference profile of a perfect matching of 𝔽_2^s.
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The Berge–Fulkerson conjecture
The Berge–Fulkerson conjecture (Fulkerson 1971). Every bridgeless cubic graph has six perfect matchings such that every edge lies in exactly two of them.
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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).
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The Bollobás–Eldridge–Catlin conjecture on graph packing (1978)
The Bollobás–Eldridge–Catlin conjecture (1978). If G_1 and G_2 are graphs on the same n vertices with maximum degrees Δ_1, Δ_2 satisfying (Δ_1 + 1)(Δ_2 + 1) ≤ n + 1, then G_1 and G_2 pack.
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The burning number conjecture
The burning number conjecture (Bonato–Janssen–Roshanbin [BJR16]). Every connected graph on n vertices can be burned in at most ⌈√n⌉ rounds: fires started at one new vertex per round, each spreading to all neighbouring vertices every round, can burn the whole graph in ⌈√n⌉ rounds.
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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.
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The circulant Hadamard conjecture
Every circulant Hadamard matrix has order at most four.
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The cup-of-coffee sequence
Conjectured to be a permutation of the positive integers.
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The Curling Number Conjecture
Starting from any nonempty finite integer sequence, repeatedly appending its curling number eventually gives a sequence with curling number 1 [Chaffin and Sloane, Conjecture 1].
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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).
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The Erdős–Moser equation
The only positive solution of S_k(m)=m^k is (k,m)=(1,3).
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The Erdős–Ulam problem
The Erdős–Ulam problem (Ulam 1945). Is there a dense subset of the plane whose points are at pairwise rational distances? Conjecturally, no such set exists.
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The first Atiyah–Sutcliffe conjecture
Atiyah–Sutcliffe Conjecture 1, stated as Conjecture 1.1 in Mazur–Petrenko: the configuration polynomials are linearly independent.
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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?
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The Goormaghtigh conjecture
The only Goormaghtigh numbers are 31 and 8191.
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The Gyárfás–Sumner conjecture
The Gyárfás–Sumner conjecture (Gyárfás 1975, Sumner 1981). For every finite tree T the class of T-free graphs (graphs with no induced subgraph isomorphic to T) is χ-bounded: there is a function f such that every T-free graph G satisfies χ(G) ≤ f(ω(G)).
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The Hasse–Weil conjecture for elliptic curves
The L-series of an elliptic curve over a number field has a meromorphic continuation to ℂ. This is the Hasse–Weil conjecture in the form stated in [Wikipedia], and a weak form of [Gross2011], Conjecture 2.7.
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The Irving–Leather conjecture on stable matchings
Thurber's §2 attributes to Irving and Leather the conjecture that their construction attains the maximum number of stable matchings at every power-of-two size.
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The length of an s-increasing sequence of r-tuples
F(n) ≤ n^3/2.
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The linear arboricity conjecture
The linear arboricity conjecture (Akiyama–Exoo–Harary [AEH81]). The edges of every graph of maximum degree Δ can be partitioned into at most ⌈(Δ + 1) / 2⌉ linear forests.
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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).
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The Manickam–Miklós–Singhi conjecture (1988)
The Manickam–Miklós–Singhi conjecture (1988). If n ≥ 4k and x_1, …, x_n are real numbers with x_1 + … + x_n = 0, then at least C(n-1, k-1) of the k-element subsets S ⊆ 1, …, n have Σ_i ∈ S x_i ≥ 0.
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The Petersen colouring conjecture (Jaeger 1988)
The Petersen colouring conjecture (Jaeger 1988). Every bridgeless cubic graph has a Petersen colouring: an edge colouring by the edges of the Petersen graph such that the three edges at each vertex are mapped onto the three edges at some vertex of the Petersen graph.
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The prime numbers
Conjecture from Thomas Ordowski (2023): log log a(n+1) - log log a(n) < 1/n for n > 0.
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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.
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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.
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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].
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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…
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The total colouring conjecture (Behzad 1965, Vizing 1968)
The total colouring conjecture (Behzad 1965, Vizing 1968). Every finite simple graph G has a total colouring with at most Δ(G) + 2 colours, i.e. χ''(G) ≤ Δ(G) + 2, where Δ(G) is the maximum degree.
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Toronto spaces
Any T2, Toronto space is discrete.
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