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On the nonnegative integer solutions to the equation F ± F = y
Journal of Number Theory ( IF 0.6 ) Pub Date : 2021-03-01 , DOI: 10.1016/j.jnt.2020.08.004
Salima Kebli , Omar Kihel , Jesse Larone , Florian Luca

Abstract In this paper, we study the solutions to the titular Diophantine equation in integers n ≥ m ≥ 0 , y ≥ 2 and a ≥ 2 . We show that there are only finitely many of them for a fixed y, and we provide a bound on the largest such solution. As an application, we find all the solutions when y ∈ [ 2 , 1000 ] . We also show that the abc-conjecture implies that there are only finitely many integer solutions ( n , m , y , a ) with min ⁡ { y , a } ≥ 2 .

中文翻译:

关于方程 F ± F = y 的非负整数解

摘要 在本文中,我们研究了整数 n ≥ m ≥ 0 , y ≥ 2 和 a ≥ 2 的名义丢番图方程的解。我们表明,对于固定的 y,它们的数量是有限的,并且我们提供了最大的此类解决方案的界限。作为一个应用,我们找到当 y ∈ [ 2 , 1000 ] 时的所有解。我们还表明 abc 猜想意味着只有有限多个整数解( n , m , y , a )且 min ⁡ { y , a } ≥ 2 。
更新日期:2021-03-01
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