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.
Let G be a graph with chromatic number aleph_1. Is it true that there is a colouring of the edges with aleph_1 many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours? A problem of Erdős, Galvin, and Hajnal.
Does there exist, for all r≥ 2, a basis A of order r (so that f_r(n)>0 for all large n) such that Σ_n≤ xf_r(n)^2 ≪ x for all x?
Is it true that in any 2-colouring of ℕ there exists an infinite set A such that all elements of A+A are the same colour? A conjecture of Owings [Ow74].
Let A be an infinite set such that there are no distinct a,b,c ∈ A such that a | (b+c) and b,c > a. Is it true that ∑_n ∈ A 1/n < ∞?
Let A ⊆ ℝ be an infinite set. Must there be a set E ⊆ ℝ of positive measure which does not contain any set of the shape a * A + b for some a,b ∈ ℝ and a ≠ 0?
Is it true that for every ε,η>0 there exists a k such that the density of n for which P(n(n+1)⋯(n+k))>n^1-ε is at least 1-η (where P(m) is the greatest prime divisor of m)?
Prove that F(n)→ ∞ as n→ ∞.
Does 1,2^3,…,N^3 contain a Sidon set of size ≫ N?
Let P_d(n) be such that in any set of n points in ℝ^d there exist at least P_d(n) many points which do not contain an isosceles triangle. Estimate P_d(n) - in particular, is it true that P_2(n)<n^1-c for some constant c>0?
Are there n such that n+2^2^k is always squarefree?
Let A⊆ [1,n) be a set of integers such that (a,b)=1 for all distinct a,b∈ A. Is it true that Σ_a∈ A1/n-a≤ Σ_p < n1/p+O(1)?
Let G be the graph with vertex set those pairs (x,y)∈ ℕ^2 with gcd(x,y)=1, in which we join two vertices if the differ in only one coordinate, and there by ± 1. Is there a path going to infinity on G, say P, such that for all (x,y)∈ P both min(x,y)>1 and at least one of x or y is composite?
Let k ≠ 0 and 3≤ d_1 < d_2 < ⋯ < d_r be integers of gcd equal to 1 such that Σ_1 ≤ i ≤ rfrac 1d_i - 1 ≥ 1. Can all sufficiently large integers be written as a sum of the shape Σ_i c_ia_i where c_i ∈ 0, 1 and a_i is divisible by d_i ^ k and has only the digits 0, 1 when written in base d_i?
Let G be a graph with n vertices such that every induced subgraph on ≥ n/2 vertices has more than n^2/50 edges. Must G contain a triangle?
We say that N is powerful if whenever p| N we also have p^2| N. Let k≥ 3. Can the product of any k consecutive positive integers ever be powerful?
In [Er80] Erdős asks whether lim_k → ∞ (W(k))^1/k = ∞
Let A ⊆ ℕ. Let B ⊆ ℕ be the set of integers which are representable in exactly one way as the sum of two elements from A. Is it true that for all ε > 0 and large N, |1,…,N ∖ B| ≫_ε N^1/2 - ε?
Let k≥3. Are there k consecutive primes in arithmetic progression?
Prove an asymptotic formula for r_k(N), the largest possible size of a subset of 1, …, N that does not contain any non-trivial k-term arithmetic progression. That is, find f_k with r_k(N) / f_k(N) → 1 as N → ∞.
Does this imply that liminf |A ∩ [1,x]|/x = 0?
Let s_1 < s_2 < ⋯ be the sequence of squarefree numbers. Is it true that, for any α≥ 0, lim_x→∞ 1/xΣ_s_n≤ x(s_n+1-s_n)^α exists?
Let F(k) be the number of solutions to 1= 1/n_1+⋯+1/n_k, where 1≤ n_1<⋯<n_k are distinct integers. Find good estimates for F(k).
Is it true that Σ_n=1^∞(-1)^nn/p_n converges, where p_n is the sequence of primes? Note: In the problem statement, p_n is the n-th prime, indexed such that p_1=2, p_2=3, …. We 0-index here to reflect how Nat.nth works.
Let A be a finite Sidon set and A+A=s_1<⋯<s_t. Is it true that 1/tΣ_1≤ i<t(s_i+1-s_i)^2 → ∞ as lvert Arvert→ ∞?
Is it true that for every k ≥ 1 we have F(N + k) ≤ F(N) + 1 for all sufficiently large N?
Does there exist a maximal Sidon set A⊂ 1,…,N of size O(N^1/3)? A question of Erdős, Sárközy, and Sós [ESS94].
Let A be an infinite B₂[2] set. Must liminf |A ∩ 1, ..., N| * N ^ (- 1 / 2) = 0?
There exists some constant c>0 such that R(C_4,K_n) ≪ n^2-c. The prize of 100 is offered in [Er78] for a proof or disproof. This problem is #17 in Ramsey Theory in the graphs problem collection.
Estimate h(n) by finding a better upper bound.
What is the limit F(N)/N as N → ∞?
Erdős Problem 17. Are there infinitely many cluster primes?
The problem is to determine the limit of the sequence F(N)/√(N) as N → ∞.
Is it true that in any finite colouring of ℕ there exist arbitrarily large finite A such that all sums and products of distinct elements in A are the same colour?
Conjecture 1. Are there infinitely many practical numbers m such that h(m) < (log log m)^O(1)? More precisely: does there exist a constant C > 0 such that for infinitely many practical numbers m, we have h(m) < (log log m)^C?
Let Q_n be the n-dimensional hypercube graph (so that Q_n has 2^n vertices and n2^n-1 edges). Prove that R(Q_n) ≪ 2^n.
Any graph on n vertices can be decomposed into O(n) many edge-disjoint cycles and edges.
What is the smallest k such that ℝ^2 can be red/blue coloured with no pair of red points unit distance apart, and no k-term arithmetic progression of blue points with distance 1?
If G is an edge-disjoint union of n copies of K_n, then is χ(G) = n?
What is the largest k such that in any permutation of ℤ there must exist a monotone k-term arithmetic progression x_1 < ⋯ < x_k? Here a permutation of ℤ is a one-sided arrangement a_1, a_2, a_3, … of the integers, i.e.
Must every permutation of ℕ, contain a monotone 4-term arithmetic progression?
Can ℕ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
Does the longest arithmetic progression of primes in 1,…,N have length o(log N)?
Is there an integer m with (m, 6) = 1 such that none of 2^k · 3^ℓ · m + 1 are prime, for any k, ℓ ≥ 0?
Let s_1 < s_2 < … be the sequence of squarefree numbers. Is it true that for any ε > 0 and large n, s_n+1 - s_n ≪_ε s_n^ε?
Is there a dense subset of ℝ^2 such that all pairwise distances are rational?
Let n ≥ 4. Are there n points in ℝ^2, no three on a line and no four on a circle, such that all pairwise distances are integers?
Can every triangle-free graph on 5n vertices be made bipartite by deleting at most n^2 edges?
A conjecture by Heath-Brown: The sum of squares of the first N gaps between consecutive primes behaves like N * (log N)^2.
Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?
Let f(n) count the number of solutions to n=p+2^k for prime p and k≥ 0. Show that f(n)=o(log n).
Let c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than c₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?
Is it true that f(N)∼ N^1/3? Originally asked to Erdős by Bose. This is discussed in problem C11 of Guy's collection [Gu04].
Let a_1 < a_2 < … be a sequence of integers such that lim_n→∞ a_n/a_n-1^2 = 1 and Σ 1/a_n ∈ ℚ. Then, for all sufficiently large n ≥ 1, a_n = a_n-1^2 - a_n-1 + 1.
Let C > 1. Does the set of integers of the form p + ⌊ C^k ⌋, for some prime p and k≥ 0, have density >0?
Let n_1 < n_2 < ⋯ be a sequence of integers such that limsup n_k/k = ∞. Is Σ_k=1^∞ 1/2^n_k transcendental?
Is Σ_n φ(n)/2^n irrational? Here φ is the Euler totient function.
Let n_1 < n_2 < … be an arbitrary sequence of integers, each with an associated residue class a_i pmodn_i. Let A be the set of integers n such that for every i either n < n_i or n not≡ a_i pmodn_i. Must the logarithmic density of A exist?
Is Σ_n=1^∞ p_n/2^n irrational? Here p_n is the n-th prime (p_1=2, p_2=3, …).
Erdős Problem 252: irrationality of the sum for a given k.
Let A⊆ℕ be an infinite set. Is Σ_n∈ A 1/2^n - 1 irrational?
Do all positive integers n have the required property?
Is a_n = 2^2^n an irrationality sequence in the above sense?
Is n! an example of an irrationality sequence?
Let N≥ 1. What is the largest t such that there are A_1,…,A_t⊆ 1,…,N with A_i∩ A_j a non-empty arithmetic progression for all i≠ j?
Is there a covering system all of whose moduli are of the form p-1 for some primes p ≥ 5?
If G is a group, can there exist an exact covering of G by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.) The conjectured answer is no: in every such exact covering, two of the subgroups have the same cardinality.
Is there an infinite Lucas sequence a_0, a_1, … where a_n+2 = a_n+1 + a_n for n ≥ 0 such that all a_k are composite, and yet no integer has a common factor with every term of the sequence?
Let k≥ 3. Is there a choice of congruence classes a_ppmodp for every prime p such that all sufficiently large integers can be written as a_p+tp for some prime p and integer t≥ k?
If A ⊆ ℕ is such that A + A contains all but finitely many integers then limsup 1_A ∗ 1_A(n) = ∞.
Let A⊆ ℕ be an infinite set and consider the following greedy algorithm for a rational x∈ (0,1): choose the minimal n∈ A not used so far such that n≥ 1/x and repeat with x replaced by x-1/n.
Let k≥2. Is it true that, for any distinct integers 1 < n_1 < ⋯ < n_k such that Σ_i=1^k 1/n_i = 1, we must have max(n_i+1 - n_i) ≥ 3?
Is it true that there are only finitely many pairs of intervals I_1, I_2 such that Σ_n_1 ∈ I_1 1/n_1 + Σ_n_2 ∈ I_2 1/n_2 ∈ ℕ?
Is it true that, for all sufficiently large k, there exist finite intervals I_1, dotsc, I_k ⊂ ℕ, distinct, not overlapping or adjacent, with |I_i| ≥ 2 for 1 ≤ i ≤ k such that 1 = Σ_i=1^k Σ_n ∈ I_i 1/n?
Let n≥ 1 and define L_n to be the least common multiple of 1,…,n and a_n by Σ_1≤ k≤ n1/k=a_n/L_n. Is it true that (a_n,L_n)=1 occurs for infinitely many n?
Let k(N) denote the smallest k such that there exists N ≤ n_1 < ⋯ < n_k with frac 1 n_1 + ... + frac 1 n_k = 1 Is it true that lim_N → ∞ k(N) - (e - 1)N = ∞?
If A ⊂ ℕ has Σ_n ∈ Afrac 1 n = ∞, then must A contain arbitrarily long arithmetic progressions?
Is it true that, for every ε > 0, h(N) = sqrt N + O_ε(N^ε)
Let f(N) be the size of the largest A⊆ 1,…,N such that there are no solutions to 1/a= 1/b+1/c with distinct a,b,c∈ A? Estimate f(N). The colouring version of this is [303], which was solved by Brown and Rödl [BrRo91].
Is it true that N(b) ≪ log log b?
Let frac a b∈ ℚ_>0 with b squarefree. Are there integers 1 < n_1 < … < n_k, each the product of two distinct primes, such that a/b=1/n_1+⋯+1/n_k?
Are there two finite set of primes P and Q such that 1 = ( Σ_p ∈ P 1/p ) ( Σ_q ∈ Q 1/q ) ? Asked by Barbeau [Ba76]. [Ba76] Barbeau, E. J., _Computer challenge corner: Problem 477: A brute force program._
Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with Σ_n ∈ A 1/n > K there exists some S ⊆ A such that 1 - exp(-(c*K)) < Σ_n ∈ S 1/n ≤ 1?
Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds: Σ_p ∈ P 1/p = 1 - 1/m?
Is there some constant c>0 such that for every n≥ 1 there exists some δ_k∈ -1,0,1 for 1≤ k≤ n with 0< lvert Σ_1≤ k≤ nδ_k/krvert < c/2^n?
What is the size of the largest A⊆1, …, N such that there is a function δ : A → -1, 1 such that Σ_n∈ A δ n/n = 0 and Σ_n∈ A'δ n/n ≠ 0 for all non-empty A'subsetneq A.
Does there exist a set A ⊆ ℕ such that |A ∩ 1, …, N| = o((log N)^2) and every sufficiently large integer can be written as p + a for some prime p and a ∈ A?
Let k≥ 3 and A⊂ ℕ be the set of kth powers. What is the order of growth of 1_A^(k)(n), i.e. the number of representations of n as the sum of k many kth powers? Does there exist some c>0 and infinitely many n such that 1_A^(k)(n) >n^c?
Is it true that f_k,k(x) ≫_ε x^1-ε for all ε>0? This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'.
Does there exist a polynomial f(x)∈ℤ[x] such that all the sums f(a)+f(b) with a < b nonnegative integers are distinct?
Writing f_k, 3(x) for the number of integers ≤ x which are the sum of three kth powers, is it true that f_k, 3(x) ≫ x ^ (3 / k)?
Does there exist A = a_1 < a_2 < ⋯ ⊂ ℕ which is a minimal basis of order 2 (i.e. every large integer is the sum of 2 elements from A, and no proper subset of A has this property), such that lim_k→∞ a_k/k^2 = c for some c ≠ 0? Erdős and Graham conjectured a negative answer to this question [ErGr80].
Erdős Problem 329. Let A ⊆ ℕ be a Sidon set. How large can lim sup_N → ∞ |A ∩ 1,…,N| / N^1/2 be?
Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0. What is the smallest possible value of lim sup n → ∞ |A ∩ 1, …, N| / N^(1/2)?
Let A⊆ ℕ and D(A) be the set of those numbers which occur infinitely often as a_1 - a_2 with a_1, a_2∈ A. What conditions on A are sufficient to ensure D(A) has bounded gaps? This is formalised here using the answer(sorry) mechanism.
Let A = 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, … be the greedy Sidon sequence: we begin with 1 and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to a + b = c + d). What is the order of growth of A?
Do infinitely many pairs (a, a+2) occur in Ulam's sequence?
For what values of 0 ≤ m < n is there a complete sequence A = a_1 ≤ a_2 ≤ ⋯ of integers such that 1. A remains complete after removing any m elements, but 2. A is not complete after removing any n elements.
For what values of t,α ∈ (0,∞) is the sequence ⌊ tα^n⌋ complete (that is, all sufficiently large integers are the sum of distinct integers of the form ⌊ tα^n⌋)?
Is there some c > 0 such that every measurable A ⊆ ℝ^2 of measure ≥ c contains the vertices of a triangle of area 1?
Let α,β∈ ℝ_>0 such that α/β is irrational. Is the multiset ⌊ α⌋,⌊ 2α⌋,⌊ 4α⌋,…∪ ⌊ β⌋,⌊ 2β⌋,⌊ 4β⌋,… complete?
