On the size of Diophantine $m$-tuples in imaginary quadratic number rings

TL;DR

This paper extends the study of Diophantine m-tuples to imaginary quadratic integer rings, proving an upper bound of 42 elements using Diophantine approximation techniques, with a simpler proof than in the positive integers case.

Contribution

It establishes an upper bound of 42 for the size of Diophantine m-tuples in imaginary quadratic rings, improving understanding of their structure in these number fields.

Findings

Proved m ≤ 42 for Diophantine m-tuples in imaginary quadratic rings.

Used a gap principle based on Diophantine approximations.

Simpler proof compared to positive integers case.

Abstract

A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that $m\leqslant 42$. Our proof is relatively simple compared to the proofs of the similar results in positive integers.