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arXiv · 2201.06232

$k$-Diophantine $m$-tuples in Finite Fields

Trajan Hammonds·Seoyoung Kim·Steven J. Miller·Arjun Nigam·Kyle Onghai·Dishant Saikia·Lalit M. Sharma
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Abstract

In this paper, we define a $k$-Diophantine $m$-tuple to be a set of $m$ positive integers such that the product of any $k$ distinct positive integers is one less than a perfect square. We study these sets in finite fields $\mathbb{F}_p$ for odd prime $p$ and guarantee the existence of a $k$-Diophantine m-tuple provided $p$ is larger than some explicit lower bound. We also give a formula for the number of 3-Diophantine triples in $\mathbb{F}_p$ as well as an asymptotic formula for the number of $k$-Diophantine $k$-tuples.

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Trajan Hammonds, Seoyoung Kim, Steven J. Miller, Arjun Nigam, Kyle Onghai, Dishant Saikia, Lalit M. Sharma. 2022-01-17. $k$-Diophantine $m$-tuples in Finite Fields. https://arxiv.org/abs/2201.06232

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