Erdős Problem #238
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?
0claims
0verified
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.
757 shown· page 6 of 16
Erdős Problem #241
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].
0claims
0verified
Erdős Problem #243
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.
0claims
0verified
Erdős Problem #244
Let C > 1. Does the set of integers of the form p + ⌊ C^k ⌋, for some prime p and k≥ 0, have density >0?
0claims
0verified
Erdős Problem #247
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?
0claims
0verified
Erdős Problem #249
Is Σ_n φ(n)/2^n irrational? Here φ is the Euler totient function.
0claims
0verified
Erdős Problem #25
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?
0claims
0verified
Erdős Problem #251
Is Σ_n=1^∞ p_n/2^n irrational? Here p_n is the n-th prime (p_1=2, p_2=3, …).
0claims
0verified
Erdős Problem #252
Erdős Problem 252: irrationality of the sum for a given k.
0claims
0verified
Erdős Problem #257
Let A⊆ℕ be an infinite set. Is Σ_n∈ A 1/2^n - 1 irrational?
0claims
0verified
Erdős Problem #261
Do all positive integers n have the required property?
0claims
0verified
Erdős Problem #263
Is a_n = 2^2^n an irrationality sequence in the above sense?
0claims
0verified
Erdős Problem #264
Is n! an example of an irrationality sequence?
0claims
0verified
Erdős Problem #272
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?
0claims
0verified
Erdős Problem #273
Is there a covering system all of whose moduli are of the form p-1 for some primes p ≥ 5?
0claims
0verified
Erdős Problem #274
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.
0claims
0verified
Erdős Problem #276
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?
0claims
0verified
Erdős Problem #279
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?
0claims
0verified
Erdős Problem #28
If A ⊆ ℕ is such that A + A contains all but finitely many integers then limsup 1_A ∗ 1_A(n) = ∞.
0claims
0verified
Erdős Problem #282
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.
0claims
0verified
Erdős Problem #287
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?
0claims
0verified
Erdős Problem #288
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?
0claims
0verified
Erdős Problem #289
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?
0claims
0verified
Erdős Problem #291
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?
0claims
0verified
Erdős Problem #295
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 = ∞?
0claims
0verified
Erdős Problem #3
If A ⊂ ℕ has Σ_n ∈ Afrac 1 n = ∞, then must A contain arbitrarily long arithmetic progressions?
0claims
0verified
Erdős Problem #30
Is it true that, for every ε > 0, h(N) = sqrt N + O_ε(N^ε)
0claims
0verified
Erdős Problem #302
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].
0claims
0verified
Erdős Problem #304
Is it true that N(b) ≪ log log b?
0claims
0verified
Erdős Problem #306
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?
0claims
0verified
Erdős Problem #307
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._
0claims
0verified
Erdős Problem #312
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?
0claims
0verified
Erdős Problem #313
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?
0claims
0verified
Erdős Problem #317
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?
0claims
0verified
Erdős Problem #319
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.
0claims
0verified
Erdős Problem #32
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?
0claims
0verified
Erdős Problem #322
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?
0claims
0verified
Erdős Problem #323
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'.
0claims
0verified
Erdős Problem #324
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?
0claims
0verified
Erdős Problem #325
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)?
0claims
0verified
Erdős Problem #326
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].
0claims
0verified
Erdős Problem #329
Erdős Problem 329. Let A ⊆ ℕ be a Sidon set. How large can lim sup_N → ∞ |A ∩ 1,…,N| / N^1/2 be?
0claims
0verified
Erdős Problem #33
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)?
0claims
0verified
Erdős Problem #332
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.
0claims
0verified
Erdős Problem #340
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?
0claims
0verified
Erdős Problem #342
Do infinitely many pairs (a, a+2) occur in Ulam's sequence?
0claims
0verified
Erdős Problem #348
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.
0claims
0verified
Erdős Problem #349
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⌋)?
0claims
0verified
Erdős Problem #352
Is there some c > 0 such that every measurable A ⊆ ℝ^2 of measure ≥ c contains the vertices of a triangle of area 1?
0claims
0verified
Erdős Problem #354
Let α,β∈ ℝ_>0 such that α/β is irrational. Is the multiset ⌊ α⌋,⌊ 2α⌋,⌊ 4α⌋,…∪ ⌊ β⌋,⌊ 2β⌋,⌊ 4β⌋,… complete?
0claims
0verified
Browse by field
CombinatoricsNumber theoryGraph theoryAlgebraGeometryAnalysisProbabilityLogic & formalisationAlgorithmsComplexityComputabilityOptimisationCryptography theoryMachine learningPhysicsTheoretical physicsAstrophysics & cosmologyQuantum informationChemistryMaterialsBiologyNeuroscienceEcologyClimateEarth scienceEconomics