Problem 394

Let $t_k(n)$ denote the least $m$ such that

n\mid m(m+1)(m+2)\cdots (m+k-1).

Is it true that

\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}

for some $c>0$ ?

Is it true that, for $k\geq 2$ ,

\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?

Source: erdosproblems.com/394

Number theory

Wiki page

Status

ClaimedProved

A full solution has been claimed but not yet accepted. The statement is true.

OPEN: the site's label (page last edited 28 October 2025). The site's proof-claims tab carries two proof claims: Snyder's full claim that both answers are yes, with $c=1/2048$ and a Lean development (claim page (Snyder, 2026)), and Pickhardt's partial lower bound $\sum_{n\le x}t_2(n)\gg x^2/((\log x)^{1/2}\log\log x)$ , which the Current assessment discusses. The derived standing, claimed and proved, departs from the label only because Snyder's full claim is pending; nothing is accepted.

Problem 394

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Proof claims

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