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On the solutions of the Diophantine equation $F_{n_1}+F_{n_2}+F_{n_3}+F_{n_4}=11^a$ | ||
| Journal of Algebra and Related Topics | ||
| دوره 13، شماره 1، مهر 2025، صفحه 73-85 اصل مقاله (167.91 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2024.24741.1540 | ||
| نویسندگان | ||
| B. Earp-Lynch1؛ S. Earp-Lynch* 1؛ A. El Baz2 | ||
| 1Department of Mathematics, Carleton University, Ottawa, Canada | ||
| 2Department of Mathematics, La Facult\'e des Sciences Dhar El Mahraz F\`{e}s, F\`es, Morocco | ||
| چکیده | ||
| 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}. | ||
| کلیدواژه‌ها | ||
| Linear recurrences؛ Exponential Diophantine problems؛ Baker's method | ||
| مراجع | ||
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[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. | ||
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