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arXiv · 2608.27356

Integers divisible by a shifted prime in a given interval

Rebecca Abi Abdallah·Valeriya Kovaleva·Jeremy Schlitt·Néo Tardy
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Abstract

In this paper we study the behaviour of $H^*(x,y,z)$, the number of integers less than $x$ possessing a divisor in the interval $(y,z]$ of the form $p-1$, where $p$ is a prime, for all values of $y = y(x)$ and $z= z(y)$. We observe multiple phase transitions at critical values of $z$ in terms of $x$ and $y$ guided largely by the anatomy of $n$. Our results generalize a result of Ford from 2017, which corresponds to the case $z = x$.

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Rebecca Abi Abdallah, Valeriya Kovaleva, Jeremy Schlitt, Néo Tardy. 2026-08-27. Integers divisible by a shifted prime in a given interval. https://arxiv.org/abs/2608.27356

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