Is there some $ϵ > 0$ such that there are infinitely many $n$ where all primes $p \leq (2 + ϵ) lo g n$ divide $1 \leq i \leq l o g n \prod (n + i)?$

Worked, still open.

number theory · solved · formalized (Lean) · 0 attempts

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write (L=\lfloor \log n\rfloor). A prime $p$ divides (\prod_{1\le i\le L}(n+i)) **iff** the interval $(n,n+L]$ contains a multiple of $p$. So your question asks whether there is some fixed (\epsilon>0) such that for infinitely many $n$, the short interval $(n,n+\log n]$ contains a multiple of **every** prime (p\le (2+\…

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · solved (literature)

theorem erdos_457 : answer(True) ↔ ∃ ε > (0 : ℝ),
    { (n : ℕ) | ∀ (p : ℕ), p ≤ (2 + ε) * Real.log n → p.Prime →
      p ∣ ∏ i ∈ Finset.Icc 1 ⌊Real.log n⌋₊, (n + i) }.Infinite

formal-conjectures/457.lean ↗

oeis

A391668 — Table read by antidiagonals. T(n,k) is the least number coprime to all numbers in [n+1, n+k].3,5,2,5,5,3,7,7,3,2,7,7,7,7,5,11,11,11,11,5,2,11,11,11,11,5,3,3,11,11,11,11,5,5,5,2,11,11,11,11,11,11,7,7,3,13,13,13,13,

links

a construction · link

#663Let

k \geq 2

and

q (n, k)

denote the least prime which does not divide

\prod_{1 \leq i \leq k} (n + i)

. Is it true that, if

k

is fixed and

n

is sufficiently large, we have

q (n, k) < (1 + o (1)) lo g n ?

A391668

status