Erdős Problem #893
Does the limit lim_n→∞ f(2n)/f(n) tend to infinity? (Other finite limits have been ruled out by [KoLu25], see below)
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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 #9
Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?
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Erdős Problem #906
Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., z | ∃ k, iteratedDeriv (n k) f z = 0 is dense.
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Erdős Problem #91
Suppose A⊂ ℝ^2 has lvert Arvert=n and minimises the number of distinct distances between points in A. Prove that for large n there are at least two (and probably many) such A which are non-similar.
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Erdős Problem #912
Prove that there exists some c>0 such that h(n) ∼ c (n/log n)^1/2 as n→ ∞.
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Erdős Problem #913
Are there infinitely many n such that if n(n + 1) = Π_i p_i^k_i is the factorisation into distinct primes then all exponents k_i are distinct?
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Erdős Problem #918
Erdős Problem #918
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Erdős Problem #930
Is it true that, for every r, there is a k such that if I_1,…,I_r are disjoint intervals of consecutive integers, all of length at least k, then Π_1≤ i≤ rΠ_m∈ I_im is not a perfect power?
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Erdős Problem #931
Let k_1 ≥ k_2 ≥ 3. Are there only finitely many n_2≥ n_1 + k_1 such that Π_1≤ i≤ k_1(n_1 + i) and Π_1≤ j≤ k_2 (n_2 + j) have the same prime factors?
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Erdős Problem #932
Let p_k denote the kth prime. For infinitely many r there are at least two integers p_r < n < p_r+1 all of whose prime factors are < p_r + 1 - p_r.
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Erdős Problem #933
If n(n+1)=2^k3^lm, where (m,6)=1, then is it true that limsup_n→ ∞ 2^k3^l/nlog n=∞?
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Erdős Problem #938
Let A=n_1 < n_2 < ⋯ be the sequence of powerful numbers (if p| n then p^2| n). Are there only finitely many three-term progressions of consecutive terms n_k,n_k+1,n_k+2?
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Erdős Problem #939
If r≥4 then can the sum of r-2 coprime r-powerful numbers ever be itself r-powerful?
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Erdős Problem #940
Let r ≥ 3. Is it true that the set of integers which are the sum of at most r r-powerful numbers has density 0?
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Erdős Problem #942
Is there some constant c > 0 such that h(n) < (log n)^c + o(1) and, for infinitely many n, h(n) > (log n)^c - o(1).
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Erdős Problem #943
Let A be the set of powerful numbers. Is is true that 1_Aast 1_A(n)=n^o(1) for every n?
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Erdős Problem #944
Let k ≥ 4 and r≥ 1. Must there exist a graph G with chromatic number k such that every vertex is critical, yet every critical set of edges has size >r?
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Erdős Problem #945
Is it true that F(x) ≤ (log x)^O(1)?
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Erdős Problem #949
Let S ⊆ ℝ be a set containing no solutions to a + b = c. Must there be a set A ⊆ ℝ ∖ S of cardinality continuum such that A + A ⊆ ℝ∖ S?
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Erdős Problem #950
Is it true that liminf f(n)=1?
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Erdős Problem #951
If 1 < a 0 < ... has property Erdos951Prop, is it true that #a i ≤ x ≤ π x?
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Erdős Problem #952
Is there an infinite sequence of distinct Gaussian primes x_1,x_2,… such that lvert x_n+1-x_nrvert ≪ 1?
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Erdős Problem #955
If A⊂ ℕ has density 0 then s^-1(A) must also have density 0. A conjecture of Erdős, Granville, Pomerance, and Spiro [EGPS90].
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Erdős Problem #959
Let A⊆ ℝ^2 be a set of size n and let d_1,…,d_k be the set of distinct distances determined by A. Let f(d) be the number of times the distance d is determined, ordered so that f(d_1)≥ f(d_2)≥ ⋯ ≥ f(d_k).
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Erdős Problem #96
If n points in ℝ^2 form a convex polygon then there are O(n) many pairs which are distance 1 apart.
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Erdős Problem #961
It is conjectured that f(k) ≪ (log k)^O(1).
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Erdős Problem #962
Main conjecture: log k(n) ≤ (log n)^(1/2 + o(1))
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Erdős Problem #968
Does the set n | u n < u (n+1) have positive lower density?
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Erdős Problem #97
Does every convex polygon have a vertex with no other 4 vertices equidistant from it?
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Erdős Problem #970
Let h(k) be Jacobsthal's function, defined to as the minimal m such that, if n has at most k prime factors, then in any set of m consecutive integers there exists an integer coprime to n. Determine the order of magnitude of h(k). In particular, is it true that h(k) ≪ k^2?
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Erdős Problem #971
Let p(a, d) be the least prime congruent to a (mod d). Does there exist a constant c > 0 such that for all large d, p(a, d) > (1 + c) φ(d) log d for ≫ φ(d) many values of a?
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Erdős Problem #972
Erdős problem 972. Let α > 1 be irrational. Are there infinitely many primes p such that ⌊ pα ⌋ is also prime?
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Erdős Problem #975
For an irreducible polynomial f ∈ ℤ[x] with f(n) ≥ 1 for sufficiently large n, does there exists a constant c = c(f) > 0 such that Σ_n ≤ x τ(f(n)) ≈ c · x log x? Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.
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Erdős Problem #978
If k>3 (and k ≠ 2^l), and for all primes p there exists n such that p^k-2nmid f(n), then are there infinitely many n for which f(n) is (k-2)-power-free?
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Erdős Problem #979
Let k ≥ 2, and let f_k(n) count the number of solutions to n = p_1^k + … + p_k^k, where the p_i are prime numbers. Is it true that limsup f_k(n) = ∞?
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Erdős Problem #98
Let h(n) be such that any n points in ℝ^2, with no three on a line and no four on a circle, determine at least h(n) distinct distances. Does h(n)/n→ ∞?
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Erdős Problem #982
If n distinct points in ℝ^2 form a convex polygon then some vertex has at least lfloorn/2⌋ different distances to other vertices.
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Erdős Problem #985
Is it true that, for every prime p, there is a prime q ≤ p which is a primitive root modulo p?
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Erdős Problem #99
For sufficiently large n, is it the case that any set of n points with minimum distance 1 that minimizes diameter must contain an equilateral triangle of side length 1?
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Erdős Problem #995
Erdős Problem 995: For every lacunary sequence (n_k) of integers and every f ∈ L^2([0,1]) with ∫_0^1 f = 0, is it true that for almost all α, Σ_k < N f(α n_k) = o (N √(loglog N))?
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Erdős Problem #996
Does there exists a positive constant C such that for all f ∈ L²[0,1] and all lacunary sequences n, if ‖f - fₖ‖₂ = O(1 / log log log k ^ C), then for almost every x, lim ∑ k ∈ Finset.range N, f (n k • x)) / N = ∫ t, f t ∂t?
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Euclid Numbers conjecture
It is not known whether there is an infinite number of prime Euclid numbers.
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Euclid-Mullin sequence
"Does the sequence ... contain every prime? ... [It] was considered by Guy and Nowakowski and later by Shanks, [Wagstaff93] computed the sequence through the 43rd term. The computational problem inherent in continuing the sequence further is the enormous size of the numbers that must be factored.
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Euler's sum of powers conjecture
Euler's sum of powers conjecture states that for integers n > 1 and k > 1, if the sum of n positive integers each raised to the k-th power equals another integer raised to the k-th power, then n ≥ k. The conjecture is known to be false for k = 4 and k = 5, but remains open for k ≥ 6.
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Expansion of (1 - x)/(1 - 2 x + 3 x^2)
It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017; Arno Berger, email, Jan 06 2017]. - N. J. A. Sloane, Feb 08 2017
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Exponentials conjectures and theorems
Four exponentials conjecture Let x_0, x_1 and y_0, y_1 be ℚ-linearly independent pairs of complex numbers, then some e^x_i y_j is transcendental.
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Factorial distance to nearest square
This sequence suggests that the distance between a factorial and the closest power is tightly bounded.
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Factorial primes
There are infinitely many factorial primes.
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Feit-Thompson conjecture on primes
There are no distinct primes p and q such that q^p - 1/q - 1 divides p^q - 1/p - 1
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Fermat-Catalan conjecture
The Fermat–Catalan conjecture states that the equation a^m + b^n = c^k has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (a^m, b^n, c^k) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying frac 1 m + frac 1 n + frac 1 k < 1.
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