Erdős Problem #1059
Are there infinitely many primes p such that p - k! is composite for each k such that 1 ≤ k! < p?
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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 #1060
The conjecture is about the function f(n) which counts the number of solutions to kσ(k)=n, where σ(k) is the sum of divisors of k. The first bound is that f(n) grows slower than any power of n^(1/loglog n). The second bound is that f(n) is at most a power of log n.
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Erdős Problem #1061
How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x? Is it true that this number is asymptotic to c * x for some constant c > 0?
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Erdős Problem #1062
Erdős asked whether the limiting density f n / n exists and, if so, whether it is irrational.
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Erdős Problem #1063
Estimate n_k by finding a better upper bound than Cambie's n_k ≤ k · lcm(1, dotsc, k-1). The comparator takes its least common multiple in ℕ and casts the result.
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Erdős Problem #1065
Are there infinitely many primes p such that p = 2^k q + 1 for some prime q and k ≥ 0? This is mentioned as B46 in Unsolved Problems in Number Theory by Richard K. Guy*
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Erdős Problem #1068
Does every graph with chromatic number aleph_1 contain a countable subgraph which is infinitely connected?
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Erdős Problem #1072
Is it true that there are infinitely many p for which f(p) = p − 1?
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Erdős Problem #1073
Is it true that A(x) ≤ x^o(1)?
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Erdős Problem #1074
Let S be the set of all m≥ 1 such that there exists a prime pnot≡ 1pmodm such that m! + 1 ≡ 0pmodp. Does lim|S∩[1, x]|/x exist?
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Erdős Problem #108
For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r) contains a subgraph of girth ≥ r and chromatic number ≥ k?
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Erdős Problem #1082
Let A⊂ ℝ^2 be a set of n points with no three on a line. Does A determine at least ⌊ n/2⌋ distinct distances?
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Erdős Problem #1083
Let d≥ 3, and let f_d(n) be the minimal m such that every set of n points in ℝ^d determines at least m distinct distances. Estimate f_d(n) - in particular, is it true that f_d(n)=n^2/d-o(1)?
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Erdős Problem #1088
Let f_d(n) be the minimal m such that any set of m points in ℝ^d contains a set of n points for which any two determined distances are distinct. Erdős Problem 1088 asks to estimate f_d(n). In particular, is it true that, for every fixed n ≥ 3, f_d(n) = 2^o(d) as d → ∞?
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Erdős Problem #1093
Are there infinitely many binomial coefficients with deficiency 1?
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Erdős Problem #1094
For all n≥ 2k the least prime factor of C(n, k) is ≤max(n/k,k), with only finitely many exceptions.
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Erdős Problem #11
Is every odd n > 1 the sum of a squarefree number and a power of 2?
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Erdős Problem #1101
1. There is NO good sequence with polynomial growth.
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Erdős Problem #1106
Let p(n) be the partition number of n and F(n) be the number of distinct prime factors of ∏_i= 1 ^ n p(n), then F(n) tends to infinity when n tends to infinity.
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Erdős Problem #1107
Let r ≥ 2. Is every large integer the sum of at most r + 1 many r-powerful numbers?
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Erdős Problem #1108
For each k ≥ 2, does the set A = Σ_n∈ Sn! : S⊂ ℕ finite of all finite sums of distinct factorials contain only finitely many k-th powers?
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Erdős Problem #1109
Let f(N) be the size of the largest subset A⊆ 1,…,N such that every n∈ A+A is squarefree. Estimate f(N). In particular, is it true that f(N)≤ N^o(1), or even f(N) ≤ (log N)^O(1)? This theorem formalizes the subpolynomial bound as f(N) = O(N^ε) for every ε > 0.
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Erdős Problem #1110
Let p>q≥ 2 be two coprime integers. We call n representable if it is the sum of integers of the form p^kq^l, none of which divide each other. If p,q≠ 2,3 then what can be said about the density of non-representable numbers?
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Erdős Problem #1113
Erdős Problem 1113. Do there exist Sierpiński numbers that possess no finite covering set of primes? Erdős and Graham [ErGr80] conjectured that the answer is yes. A negative answer would imply that there are infinitely many Fermat primes.
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Erdős Problem #1133
Let C>0. There exists ε>0 such that if n is sufficiently large the following holds. For any x_1,…,x_n∈ [-1,1] there exist y_1,…,y_n∈ [-1,1] such that, if P is a polynomial of degree m<(1+ε)n with P(x_i)=y_i for at least (1-ε)n many 1≤ i≤ n, then max_x∈ [-1,1]lvert P(x)rvert >C.
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Erdős Problem #1135
The Collatz conjecture states that for any positive integer n, there exists a natural number m such that the m-th term of the sequence is 1.
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Erdős Problem #1137
Let d_n=p_n+1-p_n, where p_n denotes the nth prime. Is it true that max_n < xd_nd_n-1/(max_n < xd_n)^2→ 0 as x→ ∞?
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Erdős Problem #1139
Let 1≤ u_1 < u_2 < ⋯ be the sequence of integers with at most 2 prime factors. Is it true that limsup_k → ∞ u_k+1-u_k/log k=∞?
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Erdős Problem #1142
Are there infinitely many n > 2 such that n - 2^k is prime for all k ≥ 1 with 2^k < n? The only known such n are 4, 7, 15, 21, 45, 75, 105 (OEIS A039669).
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Erdős Problem #1145
Let A=1≤ a_1 < a_2 < ⋯ and B=1≤ b_1 < b_2 < ⋯ be sets of integers with a_n/b_n→ 1. If A+B contains all sufficiently large positive integers then is it true that limsup 1_Aast 1_B(n)=∞? A conjecture of Erdős and Sárközy.
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Erdős Problem #1146
Is B=2^m3^n : m,n≥ 0 an essential component? In [Ru99] Ruzsa states "The simplest set with a chance to be an essential component is the collection of numbers in the form 2^m3^n and Erdős often asked whether it is an essential component or not; I do not even have a plausible guess."
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Erdős Problem #1150
Is there some constant c > 0 such that, for all large enough n and all polynomials P of degree n with coefficients in -1, 1, max_|z|=1 |P(z)| > (1 + c) √(n)?
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Erdős Problem #1159
Determine whether there exists a constant C>1 such that the following holds. Let P be a finite projective plane. Must there exist a set of points S such that 1≤ lvert S∩ ℓrvert ≤ C for all lines ℓ?
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Erdős Problem #1167
Erdős Problem 1167. Let r ≥ 2 be finite, γ ≥ 2, and λ be an infinite cardinal. Let κ_α > r be cardinals for all α < γ. Is it true that 2^λ → (κ_α + 1)_α < γ^r+1 implies λ → (κ_α)_α < γ^r? Here + means cardinal addition, so that κ_α + 1 = κ_α if κ_α is infinite. A problem of Erdős, Hajnal, and Rado.
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Erdős Problem #1175
Let κ be an uncountable cardinal. Must there exist a cardinal λ such that every graph with chromatic number λ contains a triangle-free subgraph with chromatic number κ? Shelah proved that a negative answer is consistent when κ = λ = aleph_1 (see erdos_1175.variants.aleph_one).
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Erdős Problem #1176
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.
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Erdős Problem #1192
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?
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Erdős Problem #1199
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].
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Erdős Problem #12
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 < ∞?
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Erdős Problem #120
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?
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Erdős Problem #1201
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)?
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Erdős Problem #1203
Prove that F(n)→ ∞ as n→ ∞.
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Erdős Problem #1206
Does 1,2^3,…,N^3 contain a Sidon set of size ≫ N?
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Erdős Problem #1207
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?
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Erdős Problem #1209
Are there n such that n+2^2^k is always squarefree?
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Erdős Problem #1210
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)?
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Erdős Problem #1212
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?
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Erdős Problem #124
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?
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Erdős Problem #128
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?
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Erdős Problem #137
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?
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