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On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II
International Journal of Number Theory ( IF 0.5 ) Pub Date : 2021-06-17 , DOI: 10.1142/s1793042121500871
Harold S. Erazo 1 , Carlos A. Gómez 2 , Florian Luca 3, 4, 5, 6
Affiliation  

In this paper, we show that if (Xn,Yn) is the nth solution of the Pell equation X2 dY2 = ±1 for some non-square d, then given any integer c, the equation c = Xn 2m has at most 2 integer solutions (n,m) with n 0 and m 0, except for the only pair (c,d) = (1, 2). Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences.

中文翻译:

关于 Pell 方程的 X 坐标和 2 的幂的 Pillai 问题

在本文中,我们证明如果(Xn,n)是个nPell 方程的第 th 解X2 - d2 = ±1对于一些非正方形d, 然后给定任何整数C, 方程C = Xn - 2最多有2整数解(n,)n 0 0, 除了唯一的一对(C,d) = (-1, 2). 此外,我们证明这个界限是最优的。此外,我们提出了一个关于 Pillai 问题在线性循环序列中的解数的猜想。
更新日期:2021-06-17
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