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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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1,217 Problems · 2 with reviewed evidence

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#1105The anti-Ramsey number AR(n,G)\mathrm{AR}(n,G) is the maximum possible number of colours in which the edges of KnK_n can be coloured without creating a rainbow copy of GG (i.e. one in which all edges have different colours).Source says provedFormalized
#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
#1111No statement retained — open to read what the source holdsSource says openNo formal declaration
#1112No statement retained — open to read what the source holdsSource says open (Lean)No formal declaration
#1113Erdős Problem 1113. Do there exist Sierpiński numbers that possess no finite covering set of primes?Source says openFormalized
#1114No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1115No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1116No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1117No statement retained — open to read what the source holdsSource says openNo formal declaration
#1118No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#1119Let m\mathfrak{m} be an infinite cardinal with ℵ0<m<c=2ℵ0\aleph_0 < \mathfrak{m} < \mathfrak{c} = 2^{\aleph_0}. Let {fα}\{f_\alpha\} be a family of entire functions such that, for every z0∈Cz_0 \in \mathbb{C}, there are at most m\mathfrak{m} distinct values of fα(z0)f_\alpha(z_0). Must {fα}\{f_\alpha\} have cardinality at most m\mathfrak{m}?Source says independentFormalized
#1120No statement retained — open to read what the source holdsSource says openNo formal declaration
#1121If C1,…,CnC_1,\ldots,C_n are circles in R2\mathbb{R}^2 with radii r1,…,rnr_1,\ldots,r_n such that no line disjoint from all the circles divides them into two non-empty sets then the circles can be covered by a circle of radius r=∑rir=\sum r_i.Source says proved (Lean)Formalized
#1122No statement retained — open to read what the source holdsSource says openNo formal declaration
#1123No statement retained — open to read what the source holdsSource says independentNo formal declaration
#1124No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1125Let f:R→Rf:\mathbb{R}\to \mathbb{R} be such that 2f(x)≤f(x+h)+f(x+2h)2f(x) \leq f(x+h)+f(x+2h) for every x∈Rx\in \mathbb{R} and h>0h>0. Must ff be monotonic?Source says proved (Lean)Formalized
#1126If f(x+y)=f(x)+f(y)f(x+y)=f(x)+f(y) for almost all x,y∈Rx,y\in \mathbb{R} then there exists a function gg such that g(x+y)=g(x)+g(y)g(x+y)=g(x)+g(y) for all x,y∈Rx,y\in\mathbb{R} such that f(x)=g(x)f(x)=g(x) for almost all xx.Source says proved (Lean)Formalized
#1127No statement retained — open to read what the source holdsSource says independentNo formal declaration
#1128Erdős Problem 1128 (disproved by Prikry–Mills, 1978):Source says disprovedFormalized
#1129No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1130No statement retained — open to read what the source holdsSource says provedNo formal declaration
#1131No statement retained — open to read what the source holdsSource says openNo formal declaration
#1132No statement retained — open to read what the source holdsSource says openNo formal declaration
#1133Let C>0C>0. There exists ϵ>0\epsilon>0 such that if nn is sufficiently large the following holds.Source says openFormalized
#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
#1145Let A={1≤a1<a2<⋯ }A=\{1\leq a_1 < a_2 < \cdots\} and B={1≤b1<b2<⋯ }B=\{1\leq b_1 < b_2 < \cdots\} be sets of integers with an/bn→1a_n/b_n\to 1.Source says openFormalized
#1146Is B={2m3n:m,n≥0}B=\{2^m3^n : m,n\geq 0\} an essential component?Source says openFormalized
#1147No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#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
#1150Is there some constant c>0c > 0 such that, for all large enough nn and all polynomials PP of degree nn with coefficients in {−1,1}\{-1, 1\}, max⁡∣z∣=1∣P(z)∣>(1+c)n?\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?Source says openFormalized
#1151No statement retained — open to read what the source holdsSource says openNo formal declaration
#1152No statement retained — open to read what the source holdsSource says openNo formal declaration