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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

7/12

SelectNumberQuestionOpen
#471No statement retained — open to read what the source holdsSource says provedNo formal declaration
#472No statement retained — open to read what the source holdsSource says openNo formal declaration
#473No statement retained — open to read what the source holdsSource says provedNo formal declaration
#475No statement retained — open to read what the source holdsSource says decidableNo formal declaration
#476Let A⊆FpA\subseteq \mathbb{F}_p. Let A+^A={a+b:a≠b∈A}. A\hat{+}A = \{ a+b : a\neq b \in A\}. Is it true that ∣A+^A∣≥min⁡(2∣A∣−3,p)? \lvert A\hat{+}A\rvert \geq \min(2\lvert A\rvert-3,p)? Source says proved (Lean)Formalized
#477Is there a polynomial f:Z→Zf:\mathbb{Z}\to \mathbb{Z} of degree at least 22 and a set A⊂ZA\subset \mathbb{Z} such that for any z∈Zz\in \mathbb{Z} there is exactly one a∈Aa\in A and b∈{f(n):n∈Z}b\in \{ f(n) : n\in\mathbb{Z}\} such that z=a+bz=a+b?Source says openFormalized
#478No statement retained — open to read what the source holdsSource says openNo formal declaration
#479Is it true that, for all k≠1k\neq 1, there are infinitely many nn such that 2n≡k(modn)2^n\equiv k\pmod{n}?Source says openFormalized
#480Let x1,x2,…∈[0,1]x_1,x_2,\ldots\in [0,1] be an infinite sequence. Is it true that inf⁡nlim inf⁡m→∞n∣xm+n−xm∣≤5−1/2≈0.447?\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq 5^{-1/2}\approx 0.447? A conjecture of Newman.Source says provedFormalized
#481Let a1,…,ar,b1,…,br∈Na_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N} such that ∑i1ai>1\sum_{i}\frac{1}{a_i}>1. For any finite sequence of nn (not necessarily distinct) integers A=(x1,…,xn)A=(x_1,\ldots,x_n) let T(A)T(A) denote the sequence of length rnrn given by (aixj+bi)1≤j≤n,1≤i≤r.(a_ix_j+b_i)_{1\leq j\leq n, 1\leq i\leq r}. Prove that, if A1=(1)A_1=(1) and Ai+1=T(Ai)A_{i+1}=T(A_i), then there must be some AkA_k with repeated elements.Source says proved (Lean)Formalized
#482No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#483No statement retained — open to read what the source holdsSource says openNo formal declaration
#484Prove that there exists an absolute constant c>0c>0 such that, whenever {1,…,N}\{1,\ldots,N\} is kk-coloured (and NN is large enough depending on kk) then there are at least cNcN many integers in {1,…,N}\{1,\ldots,N\} which are representable as a monochromatic sum (that is, a+ba+b where a,b∈{1,…,N}a,b\in \{1,\ldots,N\} are in the same colour class and a≠ba\neq b).Source says proved (Lean)Formalized
#486For each n∈Nn \in \mathbb{N} choose some Xn⊆Z/nZX_n \subseteq \mathbb{Z}/n\mathbb{Z}. Let B={m∈N:∀n,m≢x(modn) for all x∈Xn}B = \{m \in \mathbb{N} : \forall n, m \not\equiv x \pmod{n} \text{ for all } x \in X_n\}. Must BB have a logarithmic density?Source says openFormalized
#487Let A⊆NA\subseteq \mathbb{N} have positive density. Must there exist distinct a,b,c∈Aa,b,c\in A such that [a,b]=c[a,b]=c (where [a,b][a,b] is the least common multiple of aa and bb)?Source says proved (Lean)Formalized
#488Let AA be a finite set and B={n≥1:a∣n for some a∈A}.B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}. Is it true that, for every m>n≥max⁡(A)m>n\geq \max(A), ∣B∩[1,m]∣m<2∣B∩[1,n]∣n?\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?Source says falsifiableFormalized
#489Let A⊆NA\subseteq \mathbb{N} be a set such that ∣A∩[1,x]∣=o(x1/2)\lvert A\cap [1,x]\rvert=o(x^{1/2}). Let B={n≥1:a∤n for all a∈A}B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}. If B={b1<b2<⋯ }B=\{b_1 < b_2 < \cdots\} then is it true that lim⁡x→∞1x∑bi<x(bi+1−bi)2\lim_{x \to \infty} \frac{1}{x}\sum_{b_i < x}(b_{i+1}-b_i)^2 exists (and is finite)?Source says openFormalized
#490No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration
#491No statement retained — open to read what the source holdsSource says provedNo formal declaration
#492No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#493Does there exist a kk such that every sufficiently large integer can be written in the form ∏i=1kai−∑i=1kai\prod_{i=1}^k a_i - \sum_{i=1}^k a_i for some integers ai≥2a_i\geq 2?Source says proved (Lean)Formalized
#495Let α,β∈R\alpha,\beta \in \mathbb{R}. Is it true thatlim inf⁡n→∞n∥nα∥∥nβ∥=0\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0? This is also known as the Littlewood conjecture.Source says openFormalized
#496No statement retained — open to read what the source holdsSource says provedNo formal declaration
#520Let ff be a Rademacher multiplicative function. Does there exist some constant c>0c > 0 such that, almost surely, lim sup⁡N→∞∑m≤Nf(m)Nlog⁡log⁡N=c? \limsup_{N \to \infty} \frac{\sum_{m \leq N} f(m)}{\sqrt{N \log \log N}} = c? Source says openFormalized
#530No statement retained — open to read what the source holdsSource says openNo formal declaration
#531No statement retained — open to read what the source holdsSource says openNo formal declaration
#532If N\mathbb{N} is 2-coloured then is there some infinite set A⊆NA\subseteq \mathbb{N} such that all finite subset sums∑n∈Sn \sum_{n\in S}n(as SS ranges over all non-empty finite subsets of AA) are monochromatic?Source says proved (Lean)Formalized
#534No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#535Let r≥3r \geq 3, and let fr(N)f_r(N) denote the size of the largest subset of {1,…,N}\{1,\ldots,N\} such that no subset of size rr has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that f3(N)>Nc/log⁡log⁡Nf_3(N) > N^{c/\log\log N} for some constant c>0c > 0, and conjectured this should also be an upper bound; here we state the conjectural upper bound for all r≥3r \geq 3.Source says openFormalized
#536Let ϵ>0\epsilon>0 and NN be sufficiently large. Is it true that if A⊆{1,…,N}A\subseteq \{1,\ldots,N\} has size at least ϵN\epsilon N then there must be distinct a,b,c∈Aa,b,c\in A such that [a,b]=[b,c]=[a,c],[a, b]=[b, c]=[a, c], where [⋅,⋅][\cdot, \cdot] denotes the least common multiple?Source says openFormalized
#537Let ϵ>0\epsilon>0 and NN be sufficiently large. If A⊆{1,…,N}A\subseteq \{1,\ldots,N\} has ∣A∣≥ϵN\lvert A\rvert \geq \epsilon N then must there exist a1,a2,a3∈Aa_1,a_2,a_3\in A and distinct primes p1,p2,p3p_1,p_2,p_3 such that a1p1=a2p2=a3p3?a_1p_1=a_2p_2=a_3p_3?Source says disproved (Lean)Formalized
#538Let r≥2r\geq 2 and suppose that A⊆{1,…,N}A\subseteq\{1,\ldots,N\} is such that, for any mm, there are at most rr solutions to m=pam=pa where pp is prime and a∈Aa\in A. Give the best possible upper bound for ∑n∈A1n\sum_{n\in A}\frac{1}{n}.Source says openFormalized
#539Let h(n)h(n) be maximal such that, for any set A⊆NA\subseteq \mathbb{N} of size nn, the set{a(a,b):a,b∈A}\left\{ \frac{a}{(a,b)}: a,b\in A\right\}has size at least h(n)h(n). Estimate h(n)h(n).Source says openFormalized
#540Is it true that if A⊆Z/NZA\subseteq \mathbb{Z}/N\mathbb{Z} has size ≫N1/2\gg N^{1/2} then there exists some non-empty S⊆AS\subseteq A such that ∑n∈Sn≡0(modN)\sum_{n\in S}n\equiv 0\pmod{N}?Source says proved (Lean)Formalized
#541Let a1,…,apa_1, \dots, a_p be (not necessarily distinct) residues modulo a prime pp, such that there exists some rr so that if S⊆[p]S \subseteq [p] is non-empty and ∑i∈Sai≡0(modp)\sum_{i \in S} a_i \equiv 0 \pmod{p} then ∣S∣=r|S| = r.Source says proved (Lean)Formalized
#542No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#543No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#586No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#587Nguyen and Vu proved that ∣A∣≪N1/3(log⁡N)O(1)|A| \ll N^{1/3} (\log N)^{O(1)}.Source says solvedFormalized
#635No statement retained — open to read what the source holdsSource says openNo formal declaration
#645If ℕ is 22-coloured then there must exist a monochromatic three-term arithmetic progression x,x+d,x+2dx,x+d,x+2d such that d>xd>x.Source says proved (Lean)Formalized
#646Let p1,…,pkp_1,\ldots,p_k be distinct primes. Are there infinitely many nn such that n!n! is divisible by an even power of each of the pip_i?Source says proved (Lean)Formalized
#647Let τ(n)\tau(n) count the number of divisors of nn. Is there some n>24n > 24 such that max⁡m<n(m+τ(m))≤n+2? \max_{m < n}(m + \tau(m)) \leq n + 2? Source says verifiableFormalized
#648Let g(n)g(n) denote the largest tt such that there exist integers 2≤a1<a2<⋯<at<n2\leq a_1<a_2<\cdots <a_t <n such that P(a1)>P(a2)>⋯>P(at)P(a_1)>P(a_2)>\cdots >P(a_t) where P(m)P(m) is the greatest prime factor of mm. Estimate g(n)g(n).Source says solved (Lean)Formalized
#649Let P(m)P(m) denote the greatest prime factor of mm. Is it true that, for any two primes p,qp,q, there exists some integer nn such that P(n)=pP(n)=p and P(n+1)=qP(n+1)=q?Source says disproved (Lean)Formalized
#650Let f(m)f(m) be such that if A⊆{1,…,N}A\subseteq \{1,\ldots,N\} has ∣A∣=m\lvert A\rvert=m then every interval in [1,∞)[1,\infty) of length 2N2N contains ≥f(m)\geq f(m) many distinct integers b1,…,brb_1,\ldots,b_r where each bib_i is divisible by some ai∈Aa_i\in A, where a1,…,ara_1,\ldots,a_r are distinct.Source says solved (Lean)Formalized
#656No statement retained — open to read what the source holdsSource says provedNo formal declaration
#663No statement retained — open to read what the source holdsSource says openNo formal declaration