# On the Diophantine equation $\sum_{j=1}^kjF_j^p=F_n^q$

## TL;DR

This paper investigates a Fibonacci-based Diophantine equation involving sums of Fibonacci powers, providing complete solutions for certain exponents and conjecturing only trivial solutions exist beyond specific known cases.

## Contribution

The paper offers a complete solution for the equation when exponents are 1 or 2, and conjectures that only trivial solutions exist for other cases based on computational evidence.

## Key findings

- Complete solutions for exponents 1 and 2.
- Conjecture that only trivial solutions exist beyond specific known cases.
- Computational search supports the conjecture for p,q,k ≤ 100.

## 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 $\{1,2\}$. Based on the specific cases we could solve, and a computer search with $p,q,k\le100$ 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.