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

TL;DR

This paper investigates $k$-Diophantine $m$-tuples in finite fields, establishing existence conditions, explicit formulas for triples, and asymptotic counts for larger sets, advancing understanding of these special number sets in finite algebraic structures.

Contribution

It introduces the concept of $k$-Diophantine $m$-tuples in finite fields, provides explicit existence bounds, and derives formulas for counting such tuples.

Findings

Existence of $k$-Diophantine $m$-tuples for large primes p

Explicit formula for the number of 3-Diophantine triples in $_p$

Asymptotic formula for the number of $k$-Diophantine $k$-tuples

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.