论文标题
关于Franu \ VSić和Jadrijevi \'C的猜想：反例
On a conjecture of Franu\v sić and Jadrijevi\' c: Counter-examples
论文作者
论文摘要
令$ d \ equiv 2 \ pmod 4 $为无正方形整数，以便$ x^2 -dy^2 = - 1 $和$ x^2 -dy^2 = 6 $在整数中可解决。我们证明了在$ \ mathbb {z} [\ sqrt {d}] $中存在无限的四倍体，而当$ n \ in \ in \ in \ {（4m + 1） + 4K \ sqrt {d} $ n \ in property $ d（n）$中4K \ sqrt {d}，（4M + 3） +（4K + 2）\ sqrt {d}，（4m + 2） +（4K + 2）\ sqrt {d} \} $ for $ m，k \ in \ mathbb {z} $。结果，我们为Franu \ VSić和Jadrijevi \'C的猜想提供了很少的反面例子（请参阅猜想1.1）。
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 sić and Jadrijevi\' c (see Conjecture 1.1).
