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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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NumberQuestionOpen
#241Is it true that f(N)∼N1/3f(N)\sim N^{1/3}?openFormalized
#242For every n>2n>2 there exist distinct integers 1≤x<y<z1 ≤ x < y < z such that 4n=1x+1y+1z\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z.falsifiableFormalized
#243Let a1<a2<…a_1 < a_2 < \dots be a sequence of integers such that lim⁡n→∞anan−12=1\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1 and ∑1an∈Q\sum \frac{1}{a_n} \in \mathbb{Q}.openFormalized
#244Let C>1C > 1. Does the set of integers of the form p+⌊Ck⌋p + \lfloor C^k \rfloor, for some prime pp and k≥0k\geq 0, have density >0>0?openFormalized
#245Let A⊆NA\subseteq\mathbb{N} be an infinite set such that ∣A∩{1,...,N}∣=o(N)|A\cap \{1, ..., N\}| = o(N). Is it true that lim sup⁡N→∞∣(A+A)∩{1,...,N}∣∣A∩{1,...,N}∣≥3? \limsup_{N\to\infty}\frac{|(A + A)\cap \{1, ..., N\}|}{|A \cap \{1, ..., N\}|} \geq 3? provedFormalized
#246Let (a,b)=1(a,b)=1. The set {akbl:k,l≥0}\{a^kb^l: k,l\geq 0\} is complete - that is, every large integer is the sum of distinct integers of the form akbla^kb^l with k,l≥0k,l\geq 0.proved (Lean)Formalized
#247Let n1<n2<⋯n_1 < n_2 < \cdots be a sequence of integers such that lim sup⁡nkk=∞. \limsup \frac{n_k}{k} = \infty. Is ∑k=1∞12nk \sum_{k=1}^{\infty} \frac{1}{2^{n_k}} transcendental?openFormalized
#248Are there infinitely many nn such that ω(n+k)≪k\omega(n + k) \ll k for all k≥1k \geq 1? Here ω(n)\omega(n) is the number of distinct prime divisors of nn.provedFormalized
#249Is ∑nϕ(n)2n\sum_{n} \frac{\phi(n)}{2^n} irrational? Here ϕ\phi is the Euler totient function.openFormalized
#250Is ∑n=1∞σ(n)2n \sum_{n=1}^\infty \frac{\sigma(n)}{2^n} irrational? Here σ(n)\sigma(n) is the sum of divisors function.provedFormalized
#251Is ∑n=1∞pn2n\sum_{n=1}^\infty \frac{p_n}{2^n} irrational? Here pnp_n is the nn-th prime (p1=2,p2=3,…p_1=2, p_2=3, \dots).openFormalized
#252Erdős Problem 252: irrationality of the sum for a given kk.openFormalized
#253Let a1<a2<…a_1 < a_2 < \dotsc be an infinite sequence of positive integers such that ai+1ai→1\frac{a_{i+1}}{a_i} \to 1. If every arithmetic progression contains infinitely many integers which are the sum of distinct aia_i then every sufficiently large integer is the sum of distinct aia_i.disprovedFormalized
#254Let A⊆NA\subseteq \mathbb{N} be such that ∣A∩[1,2x]∣−∣A∩[1,x]∣→∞ as x→∞\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty and ∑n∈A{θn}=∞\sum_{n\in A} \{ \theta n\}=\infty for every θ∈(0,1)\theta\in (0,1), where {x}\{x\} is the distance of xx from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of AA.openFormalized
#255No statement retained — open to read what the source holdsprovedNo formal declaration
#256No statement retained — open to read what the source holdsopenNo formal declaration
#257Let A⊆NA\subseteq\mathbb{N} be an infinite set. Is ∑n∈A12n−1 \sum_{n\in A} \frac{1}{2^n - 1} irrational?openFormalized
#258Let an→∞a_n \to \infty be a sequence of non-zero natural numbers. Is ∑nd(n)(a1...an)\sum_n \frac{d(n)}{(a_1 ... a_n)} irrational, where d(n)d(n) is the number of divisors of nn?proved (Lean)Formalized
#259Is ∑nμ(n)2n2n\sum_{n} \mu(n)^2\frac{n}{2^n} irrational?proved (Lean)Formalized
#260Let a1<a2<⋯a_1 < a_2 < \cdots be an increasing sequence such that ann→∞\frac{a_n}{n} \to \infty. Is the sum ∑n∞an2an\sum_{n}^{\infty} \frac{a_n}{2^{a_n}} irrational?openFormalized
#261No statement retained — open to read what the source holdsopenNo formal declaration
#262No statement retained — open to read what the source holdssolvedNo formal declaration
#263Is an=22na_n = 2^{2^n} an irrationality sequence in the above sense?openFormalized
#264Is 2n2^n an example of an irrationality sequence? Kovač and Tao proved that it is not [KoTa24]openFormalized
#265No statement retained — open to read what the source holdsopenNo formal declaration
#266Let ana_n be an infinite sequence of positive integers such that ∑1an\sum \frac{1}{a_n} converges. There exists some integer t≥1t \ge 1 such that ∑1an+t\sum \frac{1}{a_n + t} is irrational.disprovedFormalized
#267Let F1=F2=1F_1=F_2=1 and Fn+1=Fn+Fn−1F_{n+1} = F_n + F_{n-1} be the Fibonacci sequence. Let n1<n2<…n_1 < n_2 < \dots be an infinite sequence with nk+1nk≥c>1\frac{n_{k+1}}{n_k} \ge c > 1. Must ∑k1Fnk\sum_k \frac 1 {F_{n_k}} be irrational?openFormalized
#268Let X be the set of points in Fin d → ℝ of the shape fun i : Fin d => ∑' n : A, (1 : ℝ) / (n + i) for some infinite subset A ⊆ ℕ such that 1 / n is summable over A. X has nonempty interior. This is proved in [KoTa24]. -proved (Lean)Formalized
#269This theorem addresses the case where the set of primes PP is infinite. In this case the sum is irrational.openFormalized
#270No statement retained — open to read what the source holdsdisprovedNo formal declaration
#271No statement retained — open to read what the source holdsopenNo formal declaration
#272Let N≥1N\geq 1. What is the largest tt such that there are A1,…,At⊆{1,…,N}A_1,\ldots,A_t\subseteq \{1,\ldots,N\} with Ai∩AjA_i\cap A_j a non-empty arithmetic progression for all i≠ji\neq j?openFormalized
#273Is there a covering system all of whose moduli are of the form p−1p-1 for some primes p≥5p \geq 5?openFormalized
#274If GG is a group, can there exist an exact covering of GG by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.)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.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?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?provedFormalized
#278No statement retained — open to read what the source holdsopenNo 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?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). 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?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.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)?proved (Lean)Formalized
#284No statement retained — open to read what the source holdsprovedNo 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 ? provedFormalized
#286No statement retained — open to read what the source holdsprovedNo 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?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}? openFormalized

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