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arXiv · 1905.01411

Optimal bounds for B\"uchi's problem in modular arithmetic II

Pablo Sáez·Xavier Vidaux·Maxim Vsemirnov
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Abstract

Given a prime $p\ge5$ and an integer $s\ge1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^s$, such that $f$ is not a square, if a sequence $(f(1),\dots,f(N))$ is a sequence of squares, then $N$ is at most $M$. We obtain this result by reducing to the case where $f$ has an invertible dominant coefficient.

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Pablo Sáez, Xavier Vidaux, Maxim Vsemirnov. 2019-05-04. Optimal bounds for B\"uchi's problem in modular arithmetic II. https://arxiv.org/abs/1905.01411

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