On the Diophantine equation ∑ j = 1 k j F j p = F n q

Gökhan Soydan; László Németh; László Szalay

Archivum Mathematicum (2018)

  • Volume: 054, Issue: 3, page 177-188
  • ISSN: 0044-8753

Abstract

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Let F n denote the n t h term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation F 1 p + 2 F 2 p + ⋯ + k F 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 { 1 , 2 } . Based on the specific cases we could solve, and a computer search with p , q , k ≤ 100 we conjecture that beside the trivial solutions only F 8 = F 1 + 2 F 2 + 3 F 3 + 4 F 4 , F 4 2 = F 1 + 2 F 2 + 3 F 3 , and F 4 3 = F 1 3 + 2 F 2 3 + 3 F 3 3 satisfy the title equation.

How to cite

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Soydan, Gökhan, Németh, László, and Szalay, László. "On the Diophantine equation $\sum _{j=1}^kjF_j^p=F_n^q$." Archivum Mathematicum 054.3 (2018): 177-188. <http://eudml.org/doc/294414>.

@article{Soydan2018,
abstract = {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.},
author = {Soydan, Gökhan, Németh, László, Szalay, László},
journal = {Archivum Mathematicum},
keywords = {Fibonacci sequence; Diophantine equation},
language = {eng},
number = {3},
pages = {177-188},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the Diophantine equation $\sum _\{j=1\}^kjF_j^p=F_n^q$},
url = {http://eudml.org/doc/294414},
volume = {054},
year = {2018},
}

TY - JOUR
AU - Soydan, Gökhan
AU - Németh, László
AU - Szalay, László
TI - On the Diophantine equation $\sum _{j=1}^kjF_j^p=F_n^q$
JO - Archivum Mathematicum
PY - 2018
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 054
IS - 3
SP - 177
EP - 188
AB - 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.
LA - eng
KW - Fibonacci sequence; Diophantine equation
UR - http://eudml.org/doc/294414
ER -

References

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  11. Pongsriiam, P., Fibonacci and Lucas numbers which are one away from their products, Fibonacci Quart. 55 (2017), 29–40. (2017) MR3620575
  12. Soydan, G., On the Diophantine equation ( x + 1 ) k + ( x + 2 ) k + ⋯ + ( l x ) k = y n , Publ. Math. Debrecen 91 (3–4) (2017), 369–382. (2017) MR3744801
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