Infinitely Many Counter Examples of a Conjecture of Franušić and Jadrijević

TL;DR

Let $d$ be a square-free integer with $d \equiv 15 \pmod{60}$ and assume the Pell-type equation $x^2 - dy^2 = -6$ is solvable in $\mathbb{Z}$. The paper proves there exist infinitely many Diophantine quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(4m+2+4k\sqrt{d})$ for even $m+k$, and uses this to show unconditionally that there are infinitely many rings $\mathbb{Z}[\sqrt{d}]$ for which the Franušić–Jadrijević conjecture fails. The construction relies on norm-$-6$ elements and norm-$1$ units in the quadratic ring, factoring $-6$ in two ways, and applying a scaling lemma to generate infinitely many quadruples; infinite families of $d$ are exhibited (e.g., $d = 360(10\alpha^2+\alpha)+15$), with explicit instances such as $d=15$ yielding $D(n)$-quadruples for $n=2(4+\sqrt{15})^{2t}$. Consequently, the conjecture fails unconditionally in these rings, expanding the landscape of $D(n)$-structures in quadratic rings and providing explicit infinite counterexamples.

Abstract

Let $d$ be a square-free integer such that $d \equiv 15 \pmod{60}$ and the Pell's equation $x^2 - dy^2 = -6$ is solvable in rational integers $x$ and $y$. In this paper, we prove that there exist infinitely many Diophantine quadruples in $\mathbb{Z}[\sqrt{d}]$ with the property $D(n)$ for certain $n$'s. As an application of it, we `unconditionally' prove the existence of infinitely many rings $\mathbb{Z}[\sqrt{d}]$ 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 $\mathcal{R}$ with the property $D(n)$ and the representability of $n$ as a difference of two squares in $\mathcal{R}$, where $\mathcal{R}$ is a commutative ring with unity.

Theorems & Definitions (16)

• Conjecture 1.1
• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof