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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

10/26

NumberQuestionOpen
#433No statement retained — open to read what the source holdsproved (Lean)No formal declaration
#434Let k≤nk \le n. What choice of A⊆{1,…,n}A\subseteq\{1, \dots, n\} (with gcd(A)=1\text{gcd}(A) = 1) of size ∣A∣=k|A| = k maximises the number of integers not representable as the sum of finitely many elements from AA (with repetitions allowed)? Is it {n,n−1,…,n−k+1}\{n, n - 1, \dots, n - k + 1\}?proved (Lean)Formalized
#435Let n∈Nn\in\mathbb{N} with n≠pkn\neq p^k for any prime pp and k≥0k\geq 0. What is the largest integer not of the form ∑1≤i<nci(ni)\sum_{1\leq i<n}c_i\binom{n}{i} where the ci≥0c_i\geq 0 are integers?proved (Lean)Formalized
#436No statement retained — open to read what the source holdsopenNo formal declaration
#437No statement retained — open to read what the source holdsprovedNo formal declaration
#438No statement retained — open to read what the source holdssolvedNo formal declaration
#439No statement retained — open to read what the source holdsprovedNo formal declaration
#440No statement retained — open to read what the source holdssolvedNo formal declaration
#441No statement retained — open to read what the source holdsdisprovedNo formal declaration
#442Let Log⁡x:=max⁡{log⁡x,1}\operatorname{Log} x := \max\{\log x, 1\}, Log⁡2x=Log⁡(Log⁡x)\operatorname{Log}_2x = \operatorname{Log} (\operatorname{Log} x), and Log⁡3x=Log⁡(Log⁡(Log⁡x)).\operatorname{Log}_3x = \operatorname{Log}(\operatorname{Log}(\operatorname{Log} x)). Is it true that if A⊆NA\subseteq\mathbb{N} is such that 1Log⁡2x∑n∈A:n≤x1n→∞ \frac{1}{\operatorname{Log}_2 x} \sum_{n\in A: n\leq x} \frac{1}{n}\to\infty then (∑n∈A:n≤x1n)2∑n,m∈A:n<m≤x1lcm⁡(n,m)→∞ \left(\sum_{n\in A: n\leq x} \frac{1}{n}\right)^2 \sum_{n, m \in A: n < m \leq x} \frac{1}{\operatorname{lcm}(n, m)}\to\infty as x→∞x\to\infty?disprovedFormalized
#443Let m,n≥1m,n\geq 1. What is #{k(m−k):1≤k≤m/2}∩{l(n−l):1≤l≤n/2}?\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}? Can it be arbitrarily large?proved (Lean)Formalized
#444No statement retained — open to read what the source holdsprovedNo formal declaration
#445Is it true that, for any c>1/2c>1/2, if pp is a sufficiently large prime then, for any n≥0n\geq 0, there exist a,b∈(n,n+pc)a,b\in(n,n+p^c) such that ab≡1(modp)ab\equiv 1\pmod{p}?openFormalized
#446No statement retained — open to read what the source holdssolvedNo formal declaration
#447How large can a union-free collection F\mathcal{F} of subsets of [n][n] be? By union-free we mean there are no solutions to A∪B=CA\cup B=C with distinct A,B,C∈FA,B,C\in \mathcal{F}. Must ∣F∣=o(2n)\lvert \mathcal{F}\rvert =o(2^n)?proved (Lean)Formalized
#448Let τ(n)\tau(n) count the divisors of nn and τ+(n)\tau^+(n) count the number of kk such that nn has a divisor in [2k,2k+1)[2^k, 2^{k+1}). Is it true that, for all ϵ>0\epsilon > 0, τ+(n)<ϵ⋅τ(n) \tau^+(n) < \epsilon \cdot \tau(n) for almost all nn?disprovedFormalized
#449No statement retained — open to read what the source holdsdisprovedNo formal declaration
#450How large must y=y(ϵ,n)y=y(\epsilon,n) be such that the number of integers in (x,x+y)(x,x+y) with a divisor in (n,2n)(n,2n) is at most ϵy\epsilon y?openFormalized
#451No statement retained — open to read what the source holdsopenNo formal declaration
#452No statement retained — open to read what the source holdsopenNo formal declaration
#453Is it true that, for all sufficiently large nn, there exists some i<ni<n such that pn2<pn+ipn−i, p_n^2 < p_{n+i}p_{n-i}, where pkp_k is the kkth prime?disproved (Lean)Formalized
#454Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?openFormalized
#455Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Must lim q n / (n ^ 2) = ∞?openFormalized
#456Is it true that mn<pnm_n<p_n for almost all nn?openFormalized
#457Is there some ϵ>0\epsilon > 0 such that there are infinitely many nn where all primes p≤(2+ϵ)log⁡np \le (2 + \epsilon) \log n divide ∏1≤i≤log⁡n(n+i)? \prod_{1 \le i \le \log n} (n + i)? proved (Lean)Formalized
#458Let lcm⁡(1,…,n)\operatorname{lcm}(1, \dots, n) denote the least common multiple of {1,…,n}\{1, \dots, n\}. Let pkp_k be the kk-th prime. Is it true that for all k≥1k \geq 1, lcm⁡(1,…,pk+1−1)<pk⋅lcm⁡(1,…,pk)\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)?falsifiableFormalized
#459Let f(u)f(u) be the largest vv such that no m∈(u,v)m\in (u,v) is composed entirely of primes dividing uvuv. Estimate f(u)f(u).solved (Lean)Formalized
#460No statement retained — open to read what the source holdsopenNo formal declaration
#461No statement retained — open to read what the source holdsopenNo formal declaration
#462No statement retained — open to read what the source holdsopenNo formal declaration
#463Is there a function ff with f(n)→∞f(n)\to\infty as n→∞n\to\infty such that, for all large nn, there is a composite number mm such that n+f(n)<m<n+p(m) n + f(n) < m < n + p(m) Here p(m)p(m) is the least prime factor of mm.openFormalized
#464Let A={n1<n2<⋯ }⊂NA=\{n_1<n_2<\cdots\}\subset \mathbb{N} be a lacunary sequence (so there exists some ϵ>0\epsilon>0 with nk+1≥(1+ϵ)nkn_{k+1}\geq (1+\epsilon)n_k for all kk). Must there exist an irrational θ\theta such that {∥θnk∥:k≥1}\{ \|\theta n_k\| : k\geq 1\} is not dense in [0,1][0,1] (where ∥x∥\| x\| is the distance to the nearest integer)?proved (Lean)Formalized
#465No statement retained — open to read what the source holdsprovedNo formal declaration
#466No statement retained — open to read what the source holdsprovedNo formal declaration
#467No statement retained — open to read what the source holdsopenNo formal declaration
#468No statement retained — open to read what the source holdsopenNo formal declaration
#469Let AA be the set of all nn such that n=d1+⋯+dkn = d_1 + ⋯ + d_k with did_i distinct proper divisors of nn, but this is not true for any m∣nm ∣ n with m<nm < n. Does: ∑n∈A1n \sum_{n ∈ A} \frac 1 n converge?proved (Lean)Formalized
#470Are there any odd weird numbers?openFormalized
#471No statement retained — open to read what the source holdsprovedNo formal declaration
#472No statement retained — open to read what the source holdsopenNo formal declaration
#473No statement retained — open to read what the source holdsprovedNo formal declaration
#474No statement retained — open to read what the source holdsnot provableNo formal declaration
#475No statement retained — open to read what the source holdsdecidableNo 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)? 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?openFormalized
#478No statement retained — open to read what the source holdsopenNo 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}?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.provedFormalized

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