On the solutions of the Diophantine equation $F_{n_1}+F_{n_2}+F_{n_3}+F_{n_4}=11^a$

Document Type : Research Paper

Authors

1 Department of Mathematics, Carleton University, Ottawa, Canada

2 Department of Mathematics‎, ‎La Facult\'e des Sciences Dhar El Mahraz F\`{e}s‎, ‎F\`es‎, ‎Morocco

Abstract

Let $F_{n}$ denote the $n$th Fibonacci number‎. ‎In this paper‎, ‎we solve the Diophantine equation $F_{n_1}+F_{n_2}+F_{n_3}+F_{n_4}=y^a$ in integers $n_1,n_2,n_3,n_4,a$ for $y=11$‎. ‎In doing so‎, ‎we disprove a recent conjecture made by Diouf and Tiebekabe in \cite{D-T}‎.

Keywords


[1] J. J. Bravo and F. Luca, On the Diophantine equation $F_n+F_m=2^a$, Quaest. Math., 39 (2016), 391–400.
[2] B. Demirtürk Bitim and R. Keskin, On solutions of the Diophantine equation $F_n-F_m=3^a$, Proc. Indian Acad. Sci. Math. Sci., 129 (2019), Article 81.
[3] I. Diouf and P. Tiebekabe, On solutions of Diophantine equation $F_{n_{1}}+F_{n_{2}}+F_{n_{3}}+F_{n_{4}}=2^{a}$, Journal of Algebra and Related Topics 9, 131–148.
[4] A. Dujella and A. Pethő, A generalization of a theorem of Baker and Davenport, A. J.
Math., 49 (1998), 291–306.
[5] F. Erduvan and R. Keskin, Nonnegative integer solutions of the equation $F_n-F_m=5^a$, Turk. J. Math., 43 (2019), 1115–1123.
[6] S. Kebli, O. Kihel, J. Larone and F. Luca, On the nonnegative integer solutions to the equation $F_n\pm F_m=y^a$, J. Number Theory, 220 (2021), 107–127.
[7] F. Luca and V. Patel, On perfect powers that are sums of two Fibonacci numbers, J. Number Theory, 189 (2018), 90–98.
[8] E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv. Ross. Akad. Nauk, Ser. Mat., 64 (2000), 125–180.
[9] Z. Şiar and R. Keskin, On the Diophantine equation$F_{n}-F_{m}=2^a$, Colloq. Math., 159 (2020), 119-126.