# Diophantine Tuples over $\mathbb{Z}_p$

## TL;DR

This paper investigates the structure and measure of Diophantine tuples over the ring of p-adic integers and their residue fields, providing new insights into their distribution and properties.

## Contribution

It introduces methods to compute and estimate the measures of Diophantine m-tuples over _p and _p, advancing understanding of their distribution in p-adic contexts.

## Key findings

- Computed measures of Diophantine tuples in _p and _p.
- Estimated the distribution of these tuples in p-adic integers.
- Provided bounds and asymptotic estimates for the measures.

## Abstract

For an element $r$ of a ring $R$, a Diophantine $D(r)$ $m$-tuple is an $m$-tuple $(a_1,a_2,\ldots,a_m)$ of elements of $R$ such that for all $i,j$ with $i\neq j$, $a_ia_j+r$ is a perfect square in $R$. In this article, we compute and estimate the measures of the sets of $D(r)$ $m$-tuples in the ring $\mathbb{Z}_p$ of $p$-adic integers, as well as its residue field $\mathbb{F}_p$.