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arXiv · 2503.13993

On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers

Temenoujka P. Peneva·Tatiana L. Todorova
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Abstract

Let $\alpha\in \mathbb{R}\setminus\mathbb{Q}$ and $\beta\in \mathbb{R}$ be given. Suppose that $a_1,\ldots,a_s$ are distinct positive integers that do not contain a reduced residue system modulo $p^2$ for any prime $p$. We prove that there exist infinitely many primes $p$ such that the inequality $||\alpha p+\beta||<p^{-1/10}$ holds and all the numbers $p+a_1,\ldots,p+a_s$ are square-free.

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Temenoujka P. Peneva, Tatiana L. Todorova. 2025-03-18. On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers. https://arxiv.org/abs/2503.13993

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