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

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

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k-Diophantine m-tuples in Finite Fields

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arxiv 2201.06232 v1 pith:ACBHFW62 submitted 2022-01-17 math.NT

k-Diophantine m-tuples in Finite Fields

Trajan Hammonds , Seoyoung Kim , Steven J. Miller , Arjun Nigam , Kyle Onghai , Dishant Saikia , Lalit M. Sharma This is my paper

classification math.NT

keywords diophantinefieldsfiniteformulaintegersmathbbnumberpositive

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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.

k-Diophantine m-tuples in Finite Fields

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