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arXiv · 2606.19863

Consecutive integers free of certain prime factors

Wouter van Doorn·Quanyu Tang
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Abstract

Let $n_k$ denote the least integer $n>2k$ such that $(n-k)(n-k+1)\cdots(n-1)$ is not divisible by any prime in the interval $(k,2k)$. Confirming a conjecture of Erd\H{o}s, we prove that, for all sufficiently large $k$, $$ n_k > e^{\frac{\log^2 k}{20 \log \log k}}. $$

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Wouter van Doorn, Quanyu Tang. 2026-06-18. Consecutive integers free of certain prime factors. https://arxiv.org/abs/2606.19863

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