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On a Diophantine equation involving powers of Fibonacci numbers. (English) Zbl 1478.11017

Let \((F_n)_{n\ge 0}\) denotes the sequence of Fibonacci numbers, given by the initial values \(F_0=0\), \(F_1=1\), and by the recurrence relation \[ F_{n+2}=F_{n+1}+F_n. \] G. Soydan et al. [Arch. Math., Brno 54, No. 3, 177–188 (2018; Zbl 1463.11047)] investigate the Diophantine equation \[ F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q \tag{1} \] in the positive integers \(k\) and \(n\), where \(p\) and \(q\) are fixed positive integers. They consider \[ F_1^p=1=F_1^q=F_2^q,\text{ and }F_1^p+2F_2^p=3=F_4 \] as trivial solutions to (1). They solve completely the equation (1) where \(p,q\in\{1,2\}\) and they have the following conjecture based upon the specific cases they could solve, and a computer search with \(p,q,k\le100\).
Conjecture 1. The non-trivial solutions to (1) are only \begin{align*} F_4^2&=\,\,9\,=F_1+2F_2+3F_3, \\ F_8&=21=F_1+2F_2+3F_3+4F_4, \\ F_4^3&=27=F_1^3+2F_2^3+3F_3^3. \end{align*}
In this paper, the authors first state some new lemmas about Fibonacci-Lucas numbers. Then skilfully combining them with the known results on Fibonacci numbers, and using primitive divisor theorem [Yu. Bilu et al., J. Reine Angew. Math. 539, 75–122 (2001; Zbl 0995.11010)] they prove that Conjecture 1 is true where \(\max\{p,q\}\le 10\). Furthermore, they use MAGMA [W. Bosma et al., J. Symb. Comput. 24, No. 3–4, 235–265 (1997; Zbl 0898.68039)] for finding all integer solutions on some elliptic curves.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11D45 Counting solutions of Diophantine equations
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References:

[1] T. Lengyel, The order of the Fibonacci and Lucas numbers, Fibonacci Quart. 33 (1995), no. 3, 234-239. · Zbl 0838.11011
[2] F. Luca and R. Oyono, An exponential Diophantine equation related to powers of two consecutive Fibonacci numbers, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 4, 45-50. · Zbl 1253.11046
[3] MAGMA Handbook, http://magma.maths.usyd.edu.au/magma/handbook/.
[4] D. Marques and A. Togbé, On the sum of powers of two consecutive Fibonacci numbers, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 10, 174-176. · Zbl 1222.11024
[5] G. Soydan, L. Németh and L. Szalay, On the Diophantine equation \(\sum^k_{j=1}jF^p_j=F^q_n\), Arch. Math. (Brno) 54 (2018), no. 3, 177-188. · Zbl 1463.11047
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