Infinitely Many Counter Examples of a Conjecture of Franušić and Jadrijević

2025/04/09 by Shubham Gupta, Gupta, Shubham
Mathematics · #11D09 #11R11 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2504.07026

openalex publication_date 2025/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let d be a square-free integer such that d ≡ 15 \pmod60 and the 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.

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