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The Diophantine equation (m2 + n2)x + (2mn)y = (m + n)2z
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2020-03-17 , DOI: 10.1142/s179304212050089x
Xiao-Hui Yan 1
Affiliation  

For fixed coprime positive integers [Formula: see text], [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text], there is a conjecture that the exponential Diophantine equation [Formula: see text] has only the positive integer solution [Formula: see text] for any positive integer [Formula: see text]. This is the analogue of Jésmanowicz conjecture. In this paper, we consider the equation [Formula: see text], where [Formula: see text] are coprime positive integers, and prove that the equation has no positive integer solution if [Formula: see text] and [Formula: see text].

中文翻译:

丢番图方程 (m2 + n2)x + (2mn)y = (m + n)2z

对于固定互质正整数[公式:见正文]、[公式:见正文]、[公式:见正文]与[公式:见正文]和[公式:见正文],有一个猜想是指数丢番图方程[公式:见正文]对于任何正整数 [公式:见正文]只有正整数解 [公式:见正文]。这与 Jésmanowicz 猜想类似。在本文中,我们考虑方程[公式:见文本],其中[公式:见文本]是互质的正整数,并证明如果[公式:见文本]和[公式:见文本]方程没有正整数解]。
更新日期:2020-03-17
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