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

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.

18 pages with 2 figure and 2 tables; comments welcome