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

8/12

Number	Question	Source says	Formal	Result here
#672	Can the product of an arithmetic progression of positive integers $n, n + d, ..., n + (k - 1)d$ of length ≥ 4, with $(n, d) = 1$ , be a perfect power?Source says verifiableFormalized	verifiable	In Lean	No Result decision
#673	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#674	Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$ ?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#675	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#676	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#677	Denote by $M(n, k)$ the least common multiple of the finite set $\{n+1, \dotsc, n+k\}$ . Is it true that for all $m \geq n + k$ , we get $M(m, k) \neq M(n, k)$ ?Source says openFormalized	open	In Lean	No Result decision
#678	Write $M(n, k)$ be the least common multiple of $\{n+1, \dotsc, n+k\}$ . Let $k$ be sufficiently large. Are there infinitely many $m, n$ with $m \geq n + k$ such that $M(n, k) > M(m, k + 1)$ ? The answer is yes, as proved in a strong form by Cambie [Ca24]. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024).Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#679	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#680	Is it true that, for all sufficiently large $n$ , there exists some $k$ such that $p(n+k)>k^2+1,$ where $p(m)$ denotes the least prime factor of $m$ ?Source says openFormalized	open	In Lean	No Result decision
#681	Erdős problem 681. Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$ , where $p(m)$ is the least prime factor of $m$ ?Source says openFormalized	open	In Lean	No Result decision
#682	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#683	There exists $c > 0$ such that $P(n, k) > \min\{n-k+1, k^{1 + c}\}$ for all $0 < k < n$ .}Source says openFormalized	open	In Lean	No Result decision
#684	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#685	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#686	Can every integer $N≥2$ be written as $N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$ for some $k≥2$ and $m≥n+k$ ?Source says openFormalized	open	In Lean	No Result decision
#687	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#688	In particular, is it true that $\epsilon_n = o(1)$ ?Source says openFormalized	open	In Lean	No Result decision
#689	Let `n` be sufficiently large. Is there some choice of congruence class `a_p` for all primes `2 ≤ p ≤ n` such that every integer in `[1,n]` satisfies at least two of the congruences `≡ a_p (mod p)`?Source says openFormalized	open	In Lean	No Result decision
#690	No statement retained — open to read what the source holdsSource says solvedNo formal declaration	solved	No formal statement	No Result decision
#691	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#692	Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$ . Is $\delta_1(n,m)$ unimodular for $m>n+1$ (i.e. increases until some $m$ then decreases thereafter)?Source says disproved (Lean)Formalized	disproved (Lean)	In Lean	No Result decision
#693	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#694	Let `Unparsed notation: f_\max(n)` be the largest $m$ such that $\phi(m) = n$ , and `Unparsed notation: f_\min(n)` be the smallest such $m$ , where $\phi$ is Euler's totient function. Investigate `Unparsed notation: \max_{n\leq x}\frac{f_\max(n)}{f_\min(n)}.` Source says solved (Lean)Formalized	solved (Lean)	In Lean	No Result decision
#695	Let $q_1 < q_2 < \cdots$ be a sequence of primes such that $q_{i + 1} \equiv 1 \pmod{q_i}$ . Is it true that $\lim_{k \to \infty} q_k^{1/k} = \infty?$ Source says openFormalized	open	In Lean	No Result decision
#696	No statement retained — open to read what the source holdsSource says solved (Lean)No formal declaration	solved (Lean)	No formal statement	No Result decision
#697	For each $m$ and $\alpha$ , the density of the set of integers which are divisible by some $d \equiv 1 \pmod{m}$ with $1 < d < \exp (m ^ \alpha)$ exists.Source says provedFormalized	proved	In Lean	No Result decision
#698	Is there some $h(n)\to \infty$ such that for all $2\leq i<j\leq n/2$ $\textrm{gcd}\left( \binom{n}{i},\binom{n}{j}\right) \geq h(n)?$ Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#699	Erdős Problem 699. Is it true that for every $1 \le i < j \le n / 2$ there exists a prime $p \ge i$ with $p \mid \gcd\big(\binom{n}{i}, \binom{n}{j}\big)$ ?Source says falsifiableFormalized	falsifiable	In Lean	No Result decision
#700	`f n` unfolds to the infimum of `fSet n`.Source says openFormalized	open	In Lean	No Result decision
#708	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#709	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#710	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#711	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#721	No statement retained — open to read what the source holdsSource says solvedNo formal declaration	solved	No formal statement	No Result decision
#726	As $n\to \infty$ ranges over integers $\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}$ ?Source says openFormalized	open	In Lean	No Result decision
#727	Let $k ≥ 2$ . Does $((n+k)!)^2∣(2n)!$ hold for infinitely many $n$ ?Source says openFormalized	open	In Lean	No Result decision
#728	Let $\varepsilon$ be sufficiently small and $C, C' > 0$ . Are there integers $a, b, n$ such that $a, b > \varepsilon n\quad a!\, b! \mid n!\, (a + b - n)!,$ and $C \log n < a + b - n < C' \log n ?$ Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#729	Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C\log n$ such that the denominator of $\frac{n!}{a!b!}$ contains only primes $\ll_C 1$ ?Source says proved (Lean)Formalized	proved (Lean)	In Lean	No Result decision
#730	Are there infinitely many pairs of integers $n < m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?Source says openFormalized	open	In Lean	No Result decision
#731	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#748	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#763	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#764	No statement retained — open to read what the source holdsSource says disprovedNo formal declaration	disproved	No formal statement	No Result decision
#768	No statement retained — open to read what the source holdsSource says openNo formal declaration	open	No formal statement	No Result decision
#769	Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$ -dimensional unit cube can be decomposed into $k$ homothetic $n$ -dimensional cubes. Give good bounds for $c(n)$ — in particular, is it true that $c(n)\gg n^n$ ?Source says openFormalized	open	In Lean	No Result decision
#770	`n + 1` is prime iff `h n = n + 1`. This is described as 'easy to see' in [Er74b].Source says openFormalized	open	In Lean	No Result decision
#771	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision
#772	No statement retained — open to read what the source holdsSource says provedNo formal declaration	proved	No formal statement	No Result decision