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Erdős Problems

1,217 source-owned questions · 604 with a formal statement · searchable by statement, number, topic and source status.

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  • Open per source608
  • Resolved per source556
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Problems

576 Problems · 1 with reviewed evidence

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SelectNumberQuestionOpen
#1099No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1100No statement retained — open to read what the source holdsSource says openNo formal declaration
#11011. There is NO good sequence with polynomial growth.Source says openFormalized
#1102If A = {a₁ < a₂ < …} has property P, then A has natural density 0. Equivalently, (a_j / j) → ∞ as j → ∞. -Source says solved (Lean)Formalized
#1103No statement retained — open to read what the source holdsSource says openNo formal declaration
#1106Let p(n)p(n) be the partition number of nn and F(n)F(n) be the number of distinct prime factors of ∏i=1np(n)∏_{i= 1} ^ {n} p(n), then F(n)F(n) tends to infinity when nn tends to infinity.Source says openFormalized
#1107Let r≥2r \ge 2. Is every large integer the sum of at most r+1r + 1 many rr-powerful numbers?Source says openFormalized
#1108For each k≥2k \geq 2, does the set A={∑n∈Sn!:S⊂N finite}A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\} of all finite sums of distinct factorials contain only finitely many kk-th powers?Source says openFormalized
#1109No statement retained — open to read what the source holdsSource says openNo formal declaration
#1110Let p>q≥2p>q\geq 2 be two coprime integers. We call nn representable if it is the sum of integers of the form pkqlp^kq^l, none of which divide each other.Source says openFormalized
#1113Erdős Problem 1113. Do there exist Sierpiński numbers that possess no finite covering set of primes?Source says openFormalized
#1122No statement retained — open to read what the source holdsSource says openNo formal declaration
#1134No statement retained — open to read what the source holdsSource says disproved (Lean)No formal declaration
#1135The Collatz conjecture states that for any positive integer nn, there exists a natural number mm such that the mm-th term of the sequence is 1.Source says openFormalized
#1136Does there exist A⊂NA\subset \mathbb{N} with lower density >1/3>1/3 such that a+b≠2ka+b\neq 2^k for any a,b∈Aa,b\in A and k≥0k\geq 0?Source says proved (Lean)Formalized
#1137Let dn=pn+1−pnd_n=p_{n+1}-p_n, where pnp_n denotes the nnth prime. Is it true that max⁡n<xdndn−1(max⁡n<xdn)2→0\frac{\max_{n < x}d_{n}d_{n-1}}{(\max_{n < x}d_n)^2}\to 0 as x→∞x\to \infty?Source says openFormalized
#1138Erdős Problem 1138. Let x/2<y<xx/2 < y < x and C>1C > 1. If d=max⁡pn<x(pn+1−pn)d = \max_{p_n < x}(p_{n+1} - p_n), where pnp_n denotes the nn-th prime, then is it true that π(y+Cd)−π(y)∼Cdlog⁡y\pi(y + Cd) - \pi(y) \sim \frac{Cd}{\log y}?Source says disproved (Lean)Formalized
#1139Let 1≤u1<u2<⋯1\leq u_1 < u_2 < \cdots be the sequence of integers with at most 22 prime factors. Is it true that lim sup⁡k→∞uk+1−uklog⁡k=∞?\limsup_{k \to \infty} \frac{u_{k+1}-u_k}{\log k}=\infty?Source says openFormalized
#1140No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#1141Are there infinitely many nn such that n−k2n-k^2 is prime for all kk with (n,k)=1(n,k)=1 and k2<nk^2 < n?Source says disproved (Lean)Formalized
#1142Are there infinitely many n>2n > 2 such that n−2kn - 2^k is prime for all k≥1k \geq 1 with 2k<n2^k < n?Source says openFormalized
#1143No statement retained — open to read what the source holdsSource says openNo formal declaration
#1144No statement retained — open to read what the source holdsSource says openNo formal declaration
#1146Is B={2m3n:m,n≥0}B=\{2^m3^n : m,n\geq 0\} an essential component?Source says openFormalized
#1148Can every large integer nn be written as n=x2+y2−z2n=x^2+y^2-z^2 with max⁡(x2,y2,z2)≤n\max(x^2,y^2,z^2)\leq n?Source says proved (Lean)Formalized
#1149No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1180No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1181No statement retained — open to read what the source holdsSource says openNo formal declaration
#1184No statement retained — open to read what the source holdsSource says openNo formal declaration
#1187No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1188Call a set of distinct integers 1<n1<⋯<nk1<n_1<\cdots<n_k with associated congruence classes ai(modni)a_i\pmod{n_i} a distinct covering system if every integer satisfies at least one of these congruences. A minimal distinct covering system is one such that no proper subset forms a covering system. Let F(x)F(x) count the number of minimal distinct covering systems with all moduli in [1,x][1,x]. Estimate F(x)F(x).Source says openFormalized
#1189No statement retained — open to read what the source holdsSource says openNo formal declaration
#1190Let ϵm=max⁡∑1ni\epsilon_m=\max \sum \frac{1}{n_i} where the maximum is taken over all finite sequences m<n1<⋯<nkm<n_1<\cdots<n_k for which there exist congruences ai(modni)a_i\pmod{n_i} such that no integer satisfies two such congruences.Source says solved (Lean)Formalized
#1195No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1196Is it true that, for any xx, if A⊂[x,∞)A\subset [x,\infty) is a primitive set of integers (so that no distinct elements of AA divide each other) thenUnparsed notation: \sum_{a\in A}\frac{1}{a\log a}&#60; 1+o(1),where the o(1)o(1) term →0\to 0 as x→∞x\to \infty? -Source says proved (Lean)Formalized
#1200No statement retained — open to read what the source holdsSource says openNo formal declaration
#1201Is it true that for every ϵ,η>0\epsilon,\eta>0 there exists a kk such that the density of nn for which P(n(n+1)⋯(n+k))>n1−ϵP(n(n+1)\cdots(n+k))>n^{1-\epsilon} is at least 1−η1-\eta (where P(m)P(m) is the greatest prime divisor of mm)?Source says openFormalized
#1202No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1203Prove that F(n)→∞F(n)\to \infty as n→∞n\to \infty.Source says openFormalized
#1204No statement retained — open to read what the source holdsSource says openNo formal declaration
#1205No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1206No statement retained — open to read what the source holdsSource says openNo formal declaration
#1209Let A={a1<a2<⋯ }A=\{a_1<a_2<\cdots\} be a sequence of integers which tends to infinity sufficiently fast. If there is an nn such that all n+akn+a_k are primes then must there exist infinitely many such nn?Source says openFormalized
#1210Let A⊆[1,n)A\subseteq [1,n) be a set of integers such that (a,b)=1(a,b)=1 for all distinct a,b∈Aa,b\in A. Is it true that ∑a∈A1n−a≤∑p<n1p+O(1)\sum_{a\in A}\frac{1}{n-a}\leq \sum_{p < n}\frac{1}{p}+O(1)?Source says openFormalized
#1211No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1212Roughness criterion (sufficiency for the anchor conditions): if a<sa < s for all ss in the leg and the leg stays below a+P−(a)a + P^-(a), then aa is coprime to the whole leg. Stated via divisibility: no prime factor of aa divides any ss with a<s<a+pa < s < a + p for all prime factors pp of aa.Source says openFormalized
#1214Let x,y≥1x,y\geq 1 be integers such that, for all n≥1n\geq 1, the set of primes dividing xn−1x^{n}-1 is equal to the set of primes dividing yn−1y^n-1. Must x=yx=y?Source says provedFormalized
#1217No statement retained — open to read what the source holdsSource says provedNo formal declaration