Erdős Problems

1,217 source-owned questions · 604 with a formal statement · searchable by statement, number, topic and source status.

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With Repository-reviewed evidence2 / 1,217

576 Problems · 1 with reviewed evidence

6/12

Number	Question	Source says	Formal	Result here
#421	Is there a sequence $1 \le d_1 < d_2 < \dots$ with density 1 such that all products $\prod_{u \le i \le v} d_i$ are distinct?Source says openFormalized	open	In Lean	No Result decision
#422	Does $f(n)$ miss infinitely many integers?Source says openFormalized	open	In Lean	No Result decision
#423	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#424	Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \ldots, a_n$ all possible values of $a_i a_j - 1$ with $i \neq j$ . Is it true that the set of integers which eventually appear has positive density?Source says openFormalized	open	In Lean	No Result decision
#425	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#427	Erdős Problem 427: is it true that, for every $n$ and $d$ , there exists $k$ such that $d \mid p_{n + 1} + \cdots + p_{n + k},$ where $p_r$ denotes the $r$ th prime?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#428	Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$ , all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$ Source says openFormalized	open	In Lean	No Result decision
#429	Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$ , then there exists some $n$ such that $n+a$ is prime for all $a\in A$ ?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#430	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#431	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#432	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#433	No statement retained — open to read what the source holdsSource says proved (Lean)No formal declaration	proved (Lean)	No formal statement	No Result decision
#434	Let $k \le n$ . What choice of $A\subseteq\{1, \dots, n\}$ (with $\text{gcd}(A) = 1$ ) of size $\|A\| = k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it $\{n, n - 1, \dots, n - k + 1\}$ ?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#435	Let $n\in\mathbb{N}$ with $n\neq p^k$ for any prime $p$ and $k\geq 0$ . What is the largest integer not of the form $\sum_{1\leq i<n}c_i\binom{n}{i}$ where the $c_i\geq 0$ are integers?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#436	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#437	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#438	No statement retained — open to read what the source holdsSource says solvedNo formal declaration	solved	No formal statement	No Result decision
#439	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#440	No statement retained — open to read what the source holdsSource says solvedNo formal declaration	solved	No formal statement	No Result decision
#441	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#442	Let $\operatorname{Log} x := \max\{\log x, 1\}$ , $\operatorname{Log}_2x = \operatorname{Log} (\operatorname{Log} x)$ , and $\operatorname{Log}_3x = \operatorname{Log}(\operatorname{Log}(\operatorname{Log} x)).$ Is it true that if $A\subseteq\mathbb{N}$ is such that $\frac{1}{\operatorname{Log}_2 x} \sum_{n\in A: n\leq x} \frac{1}{n}\to\infty$ then $\left(\sum_{n\in A: n\leq x} \frac{1}{n}\right)^2 \sum_{n, m \in A: n < m \leq x} \frac{1}{\operatorname{lcm}(n, m)}\to\infty$ as $x\to\infty$ ?Source says disprovedFormalized	disproved	In Lean	No Result decision
#443	Let $m,n\geq 1$ . What is $\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?$ Can it be arbitrarily large?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#444	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#445	Is it true that, for any $c>1/2$ , if $p$ is a sufficiently large prime then, for any $n\geq 0$ , there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$ ?Source says openFormalized	open	In Lean	No Result decision
#446	No statement retained — open to read what the source holdsSource says solvedNo formal declaration	solved	No formal statement	No Result decision
#448	Let $\tau(n)$ count the divisors of $n$ and $\tau^+(n)$ count the number of $k$ such that $n$ has a divisor in $[2^k, 2^{k+1})$ . Is it true that, for all $\epsilon > 0$ , $\tau^+(n) < \epsilon \cdot \tau(n)$ for almost all $n$ ?Source says disprovedFormalized	disproved	In Lean	No Result decision
#449	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#450	How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$ ?Source says openFormalized	open	In Lean	No Result decision
#451	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#452	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#453	Is it true that, for all sufficiently large $n$ , there exists some $i<n$ such that $p_n^2 < p_{n+i}p_{n-i},$ where $p_k$ is the $k$ th prime?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#454	Is it true that `limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤`?Source says openFormalized	open	In Lean	No Result decision
#455	Let `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) = ∞`?Source says openFormalized	open	In Lean	No Result decision
#456	Is it true that $m_n<p_n$ for almost all $n$ ?Source says openFormalized	open	In Lean	No Result decision
#457	Is there some $\epsilon > 0$ such that there are infinitely many $n$ where all primes $p \le (2 + \epsilon) \log n$ divide $\prod_{1 \le i \le \log n} (n + i)?$ Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#458	Let $\operatorname{lcm}(1, \dots, n)$ denote the least common multiple of $\{1, \dots, n\}$ . Let $p_k$ be the $k$ -th prime. Is it true that for all $k \geq 1$ , $\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)$ ?Source says falsifiableFormalized	falsifiable	In Lean	No Result decision
#459	Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$ . Estimate $f(u)$ .Source says solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#460	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#461	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#462	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#463	Is there a function $f$ with $f(n)\to\infty$ as $n\to\infty$ such that, for all large $n$ , there is a composite number $m$ such that $n + f(n) < m < n + p(m)$ Here $p(m)$ is the least prime factor of $m$ .Source says openFormalized	open	In Lean	No Result decision
#464	Let $A=\{n_1<n_2<\cdots\}\subset \mathbb{N}$ be a lacunary sequence (so there exists some $\epsilon>0$ with $n_{k+1}\geq (1+\epsilon)n_k$ for all $k$ ). Must there exist an irrational $\theta$ such that $\{ \\|\theta n_k\\| : k\geq 1\}$ is not dense in $[0,1]$ (where $\\| x\\|$ is the distance to the nearest integer)?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#465	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#466	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#467	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#468	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#469	Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct proper divisors of $n$ , but this is not true for any $m ∣ n$ with $m < n$ . Does: $\sum_{n ∈ A} \frac 1 n$ converge?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#470	Are there any odd weird numbers?Source says openFormalized	open	In Lean	No Result decision