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# Article

 Title: On the Diophantine equation $\sum _{j=1}^kjF_j^p=F_n^q$ (English) Author: Soydan, Gökhan Author: Németh, László Author: Szalay, László Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 54 Issue: 3 Year: 2018 Pages: 177-188 Summary lang: English . Category: math . Summary: Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots +kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set $\lbrace 1,2\rbrace$. Based on the specific cases we could solve, and a computer search with $p,q,k\le 100$ we conjecture that beside the trivial solutions only $F_8=F_1+2F_2+3F_3+4F_4$, $F_4^2=F_1+2F_2+3F_3$, and $F_4^3=F_1^3+2F_2^3+3F_3^3$ satisfy the title equation. (English) Keyword: Fibonacci sequence Keyword: Diophantine equation MSC: 11B39 MSC: 11D45 idZBL: Zbl 06940797 idMR: MR3847324 DOI: 10.5817/AM2018-3-177 . Date available: 2018-08-07T13:38:18Z Last updated: 2020-01-05 Stable URL: http://hdl.handle.net/10338.dmlcz/147352 . Reference:  Alvarado, S.D., Dujella, A., Luca, F.: On a conjecture regarding balancing with powers of Fibonacci numbers.Integers 12 (2012), 1127–1158. MR 3011553, 10.1515/integers-2012-0032 Reference:  Andreescu, T., Andrica, D.: Quadratic Diophantine Equations.2015, 124–126. MR 3362224 Reference:  Behera, A., Liptai, K., Panda, G.K., Szalay, L: Balancing with Fibonacci powers.Fibonacci Quart. 49 (2011), 28–33. MR 2781575 Reference:  Chaves, A.P., Marques, D., Togbé, A.: On the sum of powers of terms of a linear recurrence sequence.Bull. Braz. Math. Soc. New Series 43 (2012), 397–406. MR 3024062, 10.1007/s00574-012-0018-y Reference:  Koshy, T.: Fibonacci and Lucas Numbers with Applications.John Wiley and Sons, 2011. MR 1855020 Reference:  Luca, F., Oyono, R.: An exponential Diophantine equation related to powers of two consecutive Fibonacci numbers.Proc. Japan Acad. Ser. A 87 (2011), 45–50. MR 2803898 Reference:  Luca, F., Szalay, L.: Fibonacci diophantine triples.Glas. Mat. Ser. III 43 (63) (2008), 253–264. MR 2460699, 10.3336/gm.43.2.03 Reference:  Marques, D., Togbé, A.: On the sum of powers of two consecutive Fibonacci numbers.Proc. Japan Acad. Ser. A 86 (2010), 174–176. MR 2779831 Reference:  Panda, G.K.: Sequence balancing and cobalancing numbers.Fibonacci Quart. 45 (2007), 265–271. MR 2438198 Reference:  Pongsriiam, P.: Fibonacci and Lucas numbers associated with Brocard-Ramanujan equation.Commun. Korean Math. Soc. 91 (3) (2017), 511–522. MR 3682410 Reference:  Pongsriiam, P.: Fibonacci and Lucas numbers which are one away from their products.Fibonacci Quart. 55 (2017), 29–40. MR 3620575 Reference:  Soydan, G.: On the Diophantine equation $(x+1)^k+(x+2)^k+\dots +(lx)^k=y^n$.Publ. Math. Debrecen 91 (3–4) (2017), 369–382. MR 3744801 Reference:  Vorob’ev, N.N.: Fibonacci Numbers.Blaisdell Pub. Co. New York, 1961. Reference:  Wulczyn, G.: Problem E2158.Amer. Math. Monthly 76 (1969), 1144–1146. MR 1535701, 10.2307/2317203 .

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