Product of powers of distinct primes as sums of Fibonacci numbers

TL;DR

This paper investigates when products of powers of distinct primes can be expressed as sums of Fibonacci numbers, providing a complete characterization for certain prime pairs with multiple solutions.

Contribution

It identifies all prime pairs (q,p) with q ≤ min{1000,p} for which the Fibonacci sum equation has multiple solutions, advancing understanding of Fibonacci prime representations.

Findings

Characterized prime pairs with multiple solutions

Determined all solutions for specific prime pairs

Enhanced knowledge of Fibonacci sums involving prime powers

Abstract

Let $F_n$ be the $n$-th Fibonacci number. In this paper, we study the Diophantine equation $F_n+F_m=p^xq^y$ in nonnegative integers $n\ge m$, $x$ and $y$, where $p$ and $q$ are fixed distinct prime numbers. We determine all pairs of primes $(q,p)$ with $q\le \min\{1000,p\}$ such that the above equation has at least two solutions $(x,y)$ (and corresponding $m,n$) in positive integers.