Infinitely many counterexamples of a conjecture of Franušić and Jadrijević

Article Type

Research Article

Publication Title

Archiv Der Mathematik

Abstract

Let d be a square-free integer such that d≡15(mod60) and Pell’s equation x2-dy2=-6 is solvable in rational integers x and y. In this paper, we prove that there exist infinitely many Diophantine quadruples in Z[d] with the property D(n) for certain n’s. As an application of it, we ‘unconditionally’ prove the existence of infinitely many rings Z[d] for which the conjecture of Franušić and Jadrijević (Conjecture 1.1) does ‘not’ hold. This conjecture states a relationship between the existence of a Diophantine quadruple in R with the property D(n) and the representability of n as a difference of two squares in R, where R is a commutative ring with unity.

First Page

173

Last Page

184

DOI

10.1007/s00013-025-02144-8

Publication Date

8-1-2025

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