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An exponential Diophantine equation related to the sum of powers of two consecutive terms of a Lucas sequence and x-coordinates of Pell equations
Periodica Mathematica Hungarica ( IF 0.8 ) Pub Date : 2021-06-10 , DOI: 10.1007/s10998-021-00388-9
Harold S. Erazo , Carlos A. Gómez

For the Fibonacci sequence the identity \(F_n^2+F_{n+1}^2 = F_{2n+1}\) holds for all \(n \ge 0\). Let \({\mathcal {X}}:= (X_\ell )_{\ell \ge 1}\) be the sequence of X-coordinates of the positive integer solutions (XY) of the Pell equation \(X^2-dY^2=\pm 1\) corresponding to a nonsquare integer \(d>1\). In this paper, we investigate all positive nonsquare integers d for which there are at least two positive integers X and \(X'\) of \(\mathcal {X}\) having a representation as the sum of xth powers of two consecutive terms of a Lucas sequence. Then we solve this problem for Fibonacci numbers.



中文翻译:

与 Lucas 序列的两个连续项的幂和 Pell 方程的 x 坐标相关的指数丢番图方程

对于斐波那契数列,恒等式\(F_n^2+F_{n+1}^2 = F_{2n+1}\)对所有\(n \ge 0\) 都成立。让\({\ mathcal {X}}:=(X_ \ ELL)_ {\ ELL \ GE 1} \)是序列X的正整数解(坐标-的X,  ÿ佩尔方程的)\( X^2-dY^2=\pm 1\)对应于一个非平方整数\(d>1\)。在本文中,我们调查所有阳性非方形整数ð为其中有至少两个正整数X\(X'\)\(\ mathcal {X} \)具有表示为的总和XLucas 序列的两个连续项的 th 次幂。然后我们解决斐波那契数列的这个问题。

更新日期:2021-06-11
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