K-Diophantine m-tuples in finite fields

Abstract

In this paper, we define a [Formula: see text]-Diophantine [Formula: see text]-tuple to be a set of [Formula: see text] positive integers such that the product of any [Formula: see text] distinct positive integers is one less than a perfect square. We study these sets in finite fields [Formula: see text] for odd prime [Formula: see text] and guarantee the existence of a [Formula: see text]-Diophantine [Formula: see text]-tuple provided [Formula: see text] is larger than some explicit lower bound. We also give a formula for the number of 3-Diophantine triples in [Formula: see text] as well as an asymptotic formula for the number of [Formula: see text]-Diophantine [Formula: see text]-tuples.