关于丢番图三元组对应的椭圆曲线上整数点的杜杰拉（Dujella）猜想的反例
Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple
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中文总结 AI 辅助
本文针对丢番图三元组对应椭圆曲线上整数点的杜杰拉猜想，构造无穷多组反例，证明存在x≠-1的整数点使得三个因子均非平方数，否定了该猜想的弱化问题。
中文摘要 AI 辅助
由不同正整数组成的集合{a,b,c}若满足ab+1、ac+1、bc+1均为完全平方数，则称为丢番图三元组。杜杰拉提出猜想：对应椭圆曲线y²=(ax+1)(bx+1)(cx+1)上仅有的整数x坐标为0、d₋、d₊，当三元组包含1时还包括-1，其中d±=a+b+c+2abc±2√[(ab+1)(ac+1)(bc+1)]。该猜想的弱化问题作为杜杰拉公开问题列表中的第4.8题提出：所有x≠-1的整数点是否都使ax+1、bx+1、cx+1均为完全平方数？本文构造了无穷多个三元组{5,115,c}，存在x≠-1的整数点，使得这三个因子均为非平方数。
英文摘要
A set $\{a,b,c\}$ of distinct positive integers is called a Diophantine triple if $ab+1$, $ac+1$, and $bc+1$ are perfect squares. Dujella formulated the conjecture that the only integral $x$-coordinates on the attached elliptic curve $$y^2=(ax+1)(bx+1)(cx+1)$$ are $0,d_-,d_+$, together with $-1$ when the triple contains $1$, where $d_{\pm}=a+b+c+2abc \pm2\sqrt{(ab+1)(ac+1)(bc+1)}$. The weaker question was stated as Problem 4.8 in Dujella's list of open problems: must every integral point with $x\ne-1$ make $ax+1$, $bx+1$, and $cx+1$ all perfect squares? We construct infinitely many triples $\{5,115,c\}$ admitting an integral point with $x\ne-1$ for which all three factors are nonsquares.
发表机构
- Faculty of Mathematics, University of Rijeka（里耶卡大学数学系）
- University of Zagreb, Faculty of Electrical Engineering and Computing（萨格勒布大学电气工程和计算学院）
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