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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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#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
#322No statement retained — open to read what the source holdsSource says openNo formal declaration
#323Is it true that fk,k(x)≫ϵx1−ϵf_{k,k}(x) \gg_\epsilon x^{1-\epsilon} for all ϵ>0\epsilon>0?Source says openFormalized
#324Does there exist a polynomial f(x)∈Z[x]f(x)\in\mathbb{Z}[x] such that all the sums f(a)+f(b)f(a)+f(b) with a<ba < b nonnegative integers are distinct?Source says openFormalized
#325Writing fk,3(x)f_{k, 3}(x) for the number of integers ≤x\leq x which are the sum of three kkth powers, is it true that fk,3(x)≫x(3/k)f_{k, 3}(x) \gg x ^ (3 / k)?Source says openFormalized
#326Let A⊂NA \subset \mathbb{N} be an additive basis of order 2.Source says openFormalized
#327No statement retained — open to read what the source holdsSource says openNo formal declaration
#328Suppose A⊆NA\subseteq\mathbb{N} and C>0C>0 is such that 1A∗1A(n)≤C1_A\ast 1_A(n)\leq C for all n∈Nn\in\mathbb{N}. Can AA be partitioned into tt many subsets A1,…,AtA_1,\ldots,A_t (where t=t(C)t=t(C) depends only on CC) such that 1Ai∗1Ai(n)<C1_{A_i}\ast 1_{A_i}(n)<C for all 1≤i≤t1\leq i\leq t and n∈Nn\in \mathbb{N}?Source says disproved (Lean)Formalized
#329Erdős Problem 329. Let A ⊆ ℕ be a Sidon set. How large can lim sup_{N → ∞} |A ∩ {1,…,N}| / N^{1/2} be?Source says openFormalized
#330Does there exist a minimal basis A⊂NA \subset \mathbb{N} with positive density such that, for any n∈An \in A, the (upper) density of integers which cannot be represented without using nn is positive?Source says proved (Lean)Formalized
#331Let A,B⊆NA,B\subseteq \mathbb{N} such that for all large NN∣A∩{1,…,N}∣≫N1/2\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}and∣B∩{1,…,N}∣≫N1/2.\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}. Is it true that there are infinitely many solutions to a1−a2=b1−b2≠0a_1-a_2=b_1-b_2\neq 0 with a1,a2∈Aa_1,a_2\in A and b1,b2∈Bb_1,b_2\in B?Source says disproved (Lean)Formalized
#332Let A⊆NA\subseteq \mathbb{N} and D(A)D(A) be the set of those numbers which occur infinitely often as a1−a2a_1 - a_2 with a1,a2∈Aa_1, a_2\in A. What conditions on AA are sufficient to ensure D(A)D(A) has bounded gaps?Source says openFormalized
#333Let A⊆NA\subseteq \mathbb{N} be a set of density zero. Does there exist a BB such that A⊆B+BA\subseteq B+B and ∣B∩{1,…,N}∣=o(N1/2)\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2}) for all large NN?Source says disproved (Lean)Formalized
#334No statement retained — open to read what the source holdsSource says openNo formal declaration
#335No statement retained — open to read what the source holdsSource says openNo formal declaration
#336No statement retained — open to read what the source holdsSource says openNo formal declaration