On a conjecture of Franu\v si\'c and Jadrijevi\' c: Counter-examples

TL;DR

This paper constructs infinitely many quadruples in quadratic integer rings with specific properties, providing counterexamples to a conjecture by Franušić and Jadrijević, thus advancing understanding of solutions to certain Diophantine equations.

Contribution

It demonstrates the existence of infinitely many quadruples with property D(n) in quadratic integer rings, countering a previous conjecture.

Findings

Existence of infinitely many quadruples with property D(n) in  for specific n

Counterexamples to Franusi7 and Jadrijevi7's conjecture

Conditions on d ensuring solvability of certain equations

Abstract

Let $d\equiv 2\pmod 4$ be a square-free integer such that $x^2 - dy^2 =- 1$ and $x^2 - dy^2 = 6$ are solvable in integers. We prove the existence of infinitely many quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$ when $n \in \{(4m + 1) + 4k\sqrt{d}, (4m + 1) + (4k + 2)\sqrt{d}, (4m + 3) + 4k\sqrt{d}, (4m + 3) + (4k + 2)\sqrt{d}, (4m + 2) + (4k + 2)\sqrt{d}\}$ for $m, k \in \mathbb{Z}$. As a consequence, we provide few counter examples to a conjecture of Franu\v si\'c and Jadrijevi\' c (see Conjecture 1.1).