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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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576 Problems · 1 with reviewed evidence

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#273Is there a covering system all of whose moduli are of the form p−1p-1 for some primes p≥5p \geq 5?Source says openFormalized
#275If a finite system of rr congruences {ai(modni):1≤i≤r}\{ a_i\pmod{n_i} : 1\leq i\leq r\} (the nin_i are not necessarily distinct) covers 2r2^r consecutive integers then it covers all integers.Source says proved (Lean)Formalized
#276Is there an infinite Lucas sequence a0,a1,…a_0, a_1, \ldots where an+2=an+1+ana_{n+2} = a_{n+1} + a_n for n≥0n \ge 0 such that all aka_k are composite, and yet no integer has a common factor with every term of the sequence?Source says openFormalized
#277Is it true that, for every cc, there exists an nn such that σ(n)>cn\sigma(n)>cn but there is no covering system whose moduli all divide nn?Source says provedFormalized
#278No statement retained — open to read what the source holdsSource says openNo formal declaration
#279Let k≥3k\geq 3. Is there a choice of congruence classes ap(modp)a_p\pmod{p} for every prime pp such that all sufficiently large integers can be written as ap+tpa_p+tp for some prime pp and integer t≥kt\geq k?Source says openFormalized
#280Let n1<n2<⋯n_1<n_2<\cdots be an infinite sequence of integers with associated ak(modnk)a_k\pmod{n_k}, such that for some ϵ>0\epsilon>0 we have nk>(1+ϵ)klog⁡kn_k>(1+\epsilon)k\log k for all kk. Then #{m<nk:m≢ai(modni) for 1≤i≤k}≠o(k). \#\{ m<n_k : m\not\equiv a_i\pmod{n_i} \textrm{ for }1\leq i\leq k\}\neq o(k). Source says disproved (Lean)Formalized
#281Let n1<n2<⋯n_1<n_2<\cdots be an infinite sequence such that, for any choice of congruence classes ai(modni)a_i\pmod{n_i}, the set of integers not satisfying any of the congruences ai(modni)a_i\pmod{n_i} has density 00. Is it true that for every ϵ>0\epsilon>0 there exists some kk such that, for every choice of congruence classes aia_i, the density of integers not satisfying any of the congruences ai(modni)a_i\pmod{n_i} for 1≤i≤k1\leq i\leq k is less than ϵ\epsilon?Source says proved (Lean)Formalized
#282Let A⊆NA\subseteq \mathbb{N} be an infinite set and consider the following greedy algorithm for a rational x∈(0,1)x\in (0,1): choose the minimal n∈An\in A such that n≥1/xn\geq 1/x and repeat with xx replaced by x−1nx-\frac{1}{n}. If this terminates after finitely many steps then this produces a representation of xx as the sum of distinct unit fractions with denominators from AA.Source says openFormalized
#283Let p ⁣:Z→Zp\colon \mathbb{Z} \rightarrow \mathbb{Z} be a polynomial whose leading coefficient is positive and such that there exists no d≥2d≥2 with d∣p(n)d ∣ p(n) for all n≥1n≥1. Is it true that, for all sufficiently large mm, there exist integers 1≤n1<⋯<nk1≤n_1<\dots < n_k such that 1=1n1+⋯+1nk1=\frac{1}{n_1}+\cdots+\frac{1}{n_k} and m=p(n1)+⋯+p(nk)m=p(n_1)+\cdots+p(n_k)?Source says proved (Lean)Formalized
#284No statement retained — open to read what the source holdsSource says provedNo formal declaration
#285Let f(k)f(k) be the minimal value of nkn_k such that there exist n1<n2<⋯<nkn_1 < n_2 < \dots < n_k with 1=1n1+⋯+1nk. 1 = \frac{1}{n_1} + \cdots + \frac{1}{n_k}. Is it true that f(k)=(1+o(1))ee−1k? f(k) = (1 + o(1)) \frac{e}{e - 1} k ? Source says provedFormalized
#286No statement retained — open to read what the source holdsSource says provedNo formal declaration
#287Let k≥2k\geq2. Is it true that, for any distinct integers 1<n1<⋯<nk1 < n_1 < \cdots < n_k such that ∑i=1k1ni=1\sum_{i=1}^k \frac{1}{n_i} = 1, we must have max⁡(ni+1−ni)≥3\max(n_{i+1} - n_i) \geq 3?Source says falsifiableFormalized
#288Is it true that there are only finitely many pairs of intervals I1I_1, I2I_2 such that ∑n1∈I11n1+∑n2∈I21n2∈N? \sum_{n_1 \in I_1} \frac{1}{n_1} + \sum_{n_2 \in I_2} \frac{1}{n_2} \in \mathbb{N}? Source says openFormalized
#289Is it true that, for all sufficiently large kk, there exists finite intervals I1,…,Ik⊂NI_1, \dotsc, I_k \subset \mathbb{N} with ∣Ii∣≥2|I_i| \geq 2 for 1≤i≤k1 \leq i \leq k such that 1=∑i=1k∑n∈Ii1n. 1 = \sum_{i=1}^k \sum_{n \in I_i} \frac{1}{n}. Source says openFormalized
#290Let a≥1a\geq 1. Must there exist some b>ab>a such that ∑a≤n≤b1n=r1s1 and ∑a≤n≤b+11n=r2s2,\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and } \sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2}, with (ri,si)=1(r_i,s_i)=1 and s2<s1s_2<s_1? If so, how does this b(a)b(a) grow with aa?Source says proved (Lean)Formalized
#291Let n≥1n\geq 1 and define LnL_n to be the least common multiple of {1,…,n}\{1,\ldots,n\} and ana_n by ∑1≤k≤n1k=anLn\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.Source says openFormalized
#292No statement retained — open to read what the source holdsSource says provedNo formal declaration
#293No statement retained — open to read what the source holdsSource says openNo formal declaration
#294No statement retained — open to read what the source holdsSource says provedNo formal declaration
#295Let k(N)k(N) denote the smallest kk such that there exists N≤n1<⋯<nkN ≤ n_1 < ⋯ < n_k with 1n1+...+1nk=1\frac 1 {n_1} + ... + \frac 1 {n_k} = 1Source says openFormalized
#296Let N≥1N\geq 1 and let k(N)k(N) be maximal such that there are kk disjoint A1,…,Ak⊆{1,…,N}A_1,\ldots,A_k\subseteq \{1,\ldots,N\} with ∑n∈Ai1n=1\sum_{n\in A_i}\frac{1}{n}=1 for all ii. Estimate k(N)k(N). Is it true that k(N)=o(log⁡N)k(N)=o(\log N)?Source says proved (Lean)Formalized
#297No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#298Does every set A⊆NA \subseteq \mathbb{N} of positive density contain some finite S⊂AS \subset A such that ∑n∈S1n=1\sum_{n \in S} \frac{1}{n} = 1?Source says proved (Lean)Formalized
#299Is there an infinite sequence a1<a2<…a_1 < a_2 < \dots such that ai+1−ai=O(1)a_{i+1} - a_i = O(1) and no finite sum of 1ai\frac{1}{a_i} is equal to 1?Source says disproved (Lean)Formalized
#300No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#301No statement retained — open to read what the source holdsSource says openNo formal declaration
#302Let f(N)f(N) be the size of the largest A⊆{1,…,N}A\subseteq \{1,\ldots,N\} such that there are no solutions to 1a=1b+1c\frac{1}{a}= \frac{1}{b}+\frac{1}{c} with distinct a,b,c∈Aa,b,c\in A? Estimate f(N)f(N).Source says openFormalized
#303Is it true that in any finite colouring of the integers there exists a monochromatic solution to 1a=1b+1c\frac 1 a = \frac 1 b + \frac 1 c with distinct a,b,ca, b, c?Source says proved (Lean)Formalized
#304Is it true that N(b)≪log⁡log⁡bN(b) \ll \log \log b?Source says openFormalized
#305No statement retained — open to read what the source holdsSource says provedNo formal declaration
#306Let ab∈Q>0\frac a b\in \mathbb{Q}_{>0} with bb squarefree. Are there integers 1<n1<⋯<nk1 < n_1 < \dots < n_k, each the product of two distinct primes, such that ab=1n1+⋯+1nk\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?Source says openFormalized
#307Are there two finite set of primes PP and QQ such thatSource says verifiableFormalized
#308No statement retained — open to read what the source holdsSource says provedNo formal declaration
#309No statement retained — open to read what the source holdsSource says disprovedNo formal declaration
#310No statement retained — open to read what the source holdsSource says provedNo formal declaration
#311No statement retained — open to read what the source holdsSource says openNo formal declaration
#312Does there exist a constant c > 0 such that, for any K > 1, whenever A is a sufficiently large finite multiset of integers with ∑n∈A1/n>K\sum_{n \in A} 1/n > K there exists some S⊆AS \subseteq A such that 1−exp⁡(−(c∗K))<∑n∈S1/n≤11 - \exp(-(c*K)) < \sum_{n \in S} 1/n \le 1?Source says openFormalized
#313Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds: ∑p∈P1p=1−1m\sum_{p \in P} \frac{1}{p} = 1 - \frac{1}{m}?Source says openFormalized
#314Let n≥1n\geq 1 and let mm be minimal such that ∑n≤k≤m1k≥1\sum_{n\leq k\leq m}\frac{1}{k}\geq 1. We define ϵ(n)=∑n≤k≤m1k−1.\epsilon(n) = \sum_{n\leq k\leq m}\frac{1}{k}-1. How small can ϵ(n)\epsilon(n) be? Is it true that lim inf⁡n2ϵ(n)=0?\liminf n^2\epsilon(n)=0?Source says proved (Lean)Formalized
#315Let u1=1u_1=1 and un+1=un(un+1)u_{n+1}=u_n(u_n+1), so that ∑k≥11uk+1\sum_{k\geq 1}\frac{1}{u_k+1} and uk=⌊c02k+1⌋u_k=\lfloor c_0^{2^k}+1\rfloor for k≥1k\geq 1, where c0=lim⁡un1/2n=1.264085⋯ .c_0=\lim u_n^{1/2^n}=1.264085\cdots. Let a1<a2<⋯a_1<a_2<\cdots be any other sequence with ∑1ak=1\sum \frac{1}{a_k}=1. Is it true that lim inf⁡an1/2n<c0=1.264085⋯?\liminf a_n^{1/2^n}<c_0=1.264085\cdots?Source says proved (Lean)Formalized
#316Is it true that if A⊆N∖{1}A \subseteq \mathbb{N}\setminus\{1\} is a finite set with ∑n∈A1n<2\sum_{n \in A} \frac{1}{n} < 2 then there is a partition A=A1⊔A2A=A_1 \sqcup A_2 such that ∑n∈Ai1n<1\sum_{n \in A_i} \frac{1}{n} < 1 for i=1,2i=1,2?Source says disproved (Lean)Formalized
#317Inequality in erdos_317.variants.claim2 is obvious, the problem is strict inequality.Source says openFormalized
#318There exists a set A with positive density that does not have property P₁. #TODO: prove this lemma by assuming erdos_318.contain_single_even.Source says solvedFormalized
#319What is the size of the largest A⊆{1,…,N}A\subseteq\{1, \dots, N\} such that there is a function δ:A→{−1,1}\delta : A \to \{-1, 1\} such that ∑n∈Aδnn=0 \sum_{n\in A} \frac{\delta n}{n} = 0 and ∑n∈A′δnn≠0 \sum_{n\in A'}\frac{\delta n}{n} \neq 0 for all non-empty A′⊊AA'\subsetneq A.Source says openFormalized
#320No statement retained — open to read what the source holdsSource says solvedNo formal declaration
#321Let R(N)R(N) be the size of the largest A⊆{1,...,N}A\subseteq\{1, ..., N\} such that all sums ∑n∈S1n\sum_{n\in S} \frac{1}{n} are distinct for S⊆AS\subseteq A. What is R(N)R(N)?Source says solvedFormalizedResult here accepted