Erdős Problem #357
Let f(n) be the maximal k such that there exist integers 1 ≤ a_1 < dotsc < a_k ≤ n such that all sums of the shape Σ_u ≤ i ≤ v a_i are distinct. Is f(n)=o(n)?
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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.
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Erdős Problem #359
Let a_1< a_2 < ⋯ be an infinite sequence of integers such that a_1=1 and a_i+1 is the least integer which is not a sum of consecutive earlier a_js. Show that a_k / k → ∞.
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Erdős Problem #361
Let c > 0 and n be some large integer. What is the size of the largest set A ⊆ 1, …, ⌊ c n ⌋ such that n is not a sum of a subset of A? Does this depend on n in an irregular way?
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Erdős Problem #364
There is no consecutive triple of powerful numbers.
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Erdős Problem #366
Are there any 2-full n such that n+1 is 3-full?
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Erdős Problem #367
Let B_2(n) be the 2-full part of n (that is, B_2(n)=n/n' where n' is the product of all primes that divide n exactly once). Is it true that, for every fixed k ≥ 1, Π_n ≤ m < n+k B_2(m) ≪ n^2+o(1)?
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Erdős Problem #371
Let P(n) denote the largest prime factor of n. Show that the set of n with P(n+1) > P(n) has density 1/2.
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Erdős Problem #373
Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.
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Erdős Problem #375
Is Erdos375Prop true?
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Erdős Problem #376
Are there infinitely many n such that 2nchoose n is coprime to 105?
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Erdős Problem #377
Is there some absolute constant C > 0 such that Σ_p ≤ n 1_pnmid 2n choose n1/p ≤ C for all n?
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Erdős Problem #383
Is it true that for every k there are infinitely many primes p such that the largest prime divisor of Π_i = 0^k (p ^ 2 + i) is p?
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Erdős Problem #385
Let F(n) := maxm + p(m) | textrmm < n composite where p(m) is the least prime divisor of m. Is it true that F(n)>n for all sufficiently large n?
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Erdős Problem #386
Let 2 ≤ k ≤ n - 2. Can C(n, k) be the product of consecutive primes infinitely often? Here k may vary with n: the question asks for infinitely many admissible binomial coefficients, not for a single k that works infinitely often.
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Erdős Problem #389
Is it true that for every n ≥ 1 there is a k such that n(n + 1) ⋯ (n + k - 1) | (n + k) ⋯ (n + 2k - 1)?
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Erdős Problem #39
Is there an infinite Sidon set A⊂ ℕ such that lvert A∩ 1…,Nrvert ≫_ε N^1/2-ε for all ε > 0?
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Erdős Problem #390
Does there exists a constant c such that f n - 2 n ~ c (n / log n)?
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Erdős Problem #396
Is it true that for every k there exists n such that Π_0≤ i≤ k(n-i) | C(2n, n)?
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Erdős Problem #398
Brocard's Problem Does n! + 1 = m^2 have integer solutions other than n = 4, 5, 7?
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Erdős Problem #40
For what functions g(N) → ∞ is it true that lvert A∩ 1,…,Nrvert ≫ N^1/2/g(N) implies limsup 1_Aast 1_A(n)=∞?
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Erdős Problem #400
Can one show that Σ_n≤ xg_k(n) ∼ c_k xlog x for some constant c_k?
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Erdős Problem #406
Is it true that there are only finitely many powers of 2 which have only the digits 0 and 1 when written in base 3?
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Erdős Problem #409
Erdős Problem #409
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Erdős Problem #41
Let A ⊂ ℕ be an infinite set such that the triple sums a+b+c are all distinct for a,b,c ∈ A (aside from the trivial coincidences). Is it true that liminf_N → ∞ fraclvert A ∩ 1,…,NrvertN^1/3=0?
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Erdős Problem #410
Let σ_1(n) = σ(n), the sum of divisors function, and σ_k(n) = σ(σ_k-1(n)). Is it true that lim_k → ∞ σ_k(n)^frac 1 k = ∞?
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Erdős Problem #412
Let σ_1(n)=σ(n), the sum of divisors function, and σ_k(n) = σ(σ_k-1(n)). Is it true that, for every m, n ≥ 2, there exist some i, j such that σ_i(m) = σ_j(n)?
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Erdős Problem #413
Are there infinitely many barriers for ω?
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Erdős Problem #414
Let h_1(n) = h(n) and h_k(n) = h(h_k-1(n)). Is it true, for any m,n, there exist i and j such that h_i(m) = h_j(n)?
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Erdős Problem #416
Let V(x) count the number of n≤x such that ϕ(m)=n is solvable. Does V(2x)/V(x)→2 ?
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Erdős Problem #417
LetV'(x)=\#φ(m) : 1≤ m≤ xandV(x)=\#φ(m) ≤ x : 1≤ m. Does lim V(x)/V'(x) exist?
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Erdős Problem #422
Let f(1) = f(2) = 1 and for n > 2 f(n) = f(n - f(n - 1)) + f(n - f(n - 2)). Does f(n) miss infinitely many integers?
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Erdős Problem #423
Erdős Problem 423 [Er77c, p.71; ErGr80, p.83]: Let a(1) = 1, a(2) = 2, and for k ≥ 3 let a(k) be the least integer greater than a(k-1) that is a sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence? It seems likely that a_n = n + o(n).
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Erdős Problem #428
Is there a set A⊆ ℕ such that, for infinitely many n, all of n-a are prime for all a∈ A with 0 < a < n and liminflvert A∩ [1,x]rvert/π(x)>0?
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Erdős Problem #431
Are there two infinite sets A and B such that A+B agrees with the primes up to finitely many exceptions?
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Erdős Problem #44
Erdős Problem 44: Let N ≥ 1 and A ⊆ 1,…,N be a Sidon set. Is it true that, for any ε > 0, there exist M = M(ε) and B ⊆ N+1,…,M such that A ∪ B ⊆ 1,…,M is a Sidon set of size at least (1−ε)M^1/2?
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Erdős Problem #445
Is it true that, for any c>1/2, if p is a sufficiently large prime then, for any n≥ 0, there exist a,b∈(n,n+p^c) such that ab≡ 1pmodp? This is discussed in this MathOverflow question [MathOverflow].
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Erdős Problem #450
How large must y=y(ε,n) be such that the number of integers in (x,x+y) with a divisor in (n,2n) is at most ε y? The bound is required for every x and every window length at least y, and y(ε,n) is the least such threshold (or ∞ if there is none).
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Erdős Problem #452
Determine the largest length of an interval in [x,2x] on which ω(n) > loglog n everywhere.
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Erdős Problem #454
Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?
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Erdős Problem #455
Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Must lim q n / (n ^ 2) = ∞?
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Erdős Problem #456
Is it true that m_n<p_n for almost all n?
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Erdős Problem #458
Let lcm(1, …, n) denote the least common multiple of 1, …, n. Let p_k be the k-th prime. Is it true that for all k ≥ 1, lcm(1, …, p_k+1-1) < p_k · lcm(1, …, p_k)?
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Erdős Problem #462
Let p(n) denote the least prime factor of n. Is there a constant C>0 such that Σ_x≤ n≤ x+C√(x)(log x)^2p(n)/n≫ 1 for all sufficiently large x?
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Erdős Problem #463
Is there a function f with f(n)→∞ as n→∞ such that, for all large n, there is a composite number m such that n + f(n) < m < n + p(m) Here p(m) is the least prime factor of m.
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Erdős Problem #470
Are there any odd weird numbers?
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Erdős Problem #478
Let p be a prime and A_p = k! pmodp : 1≤ k<p. Is it true that lvert A_prvert ∼ (1-1/e)p?
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Erdős Problem #479
Is it true that, for every integer k≠ 1, there are infinitely many n such that 2^n≡ kpmodn?
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Erdős Problem #488
Let A be a finite set and B= n ≥ 1 : a| ntextrm for some a∈ A. Is it true that, for every m>n≥ max(A), lvert B∩ [1,m]rvert /m< 2lvert B∩ [1,n]rvert/n?
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Erdős Problem #495
Let α,β ∈ ℝ. Is it true thatliminf_n→ ∞ n ‖ nα ‖ ‖ nβ‖ =0? This is also known as the Littlewood conjecture.
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Erdős Problem #5
Let C≥ 0. Is there an infinite sequence of n_i such that lim_i→ inftyp_n_i+1-p_n_i/log n_i=C? We formalise "an infinite sequence of n_i" as a strictly monotone sequence of indices n : ℕ → ℕ.
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