On the multiplicity in Pillai's problem with Fibonacci numbers and powers of a fixed prime

TL;DR

This paper investigates the representations of integers as Fibonacci numbers minus prime powers, establishing an upper bound of four on the number of such representations for any integer.

Contribution

It provides a new upper bound on the multiplicity of representations of integers as Fibonacci numbers minus prime powers, extending understanding of Pillai's problem in this context.

Findings

Maximum of four representations for any integer c

Bound applies universally for all primes p

Advances knowledge on Fibonacci and prime power differences

Abstract

Let $ \{F_n\}_{n\ge 0} $ be the sequence of Fibonacci numbers and let $p$ be a prime. For an integer $c$ we write $m_{F,p}(c)$ for the number of distinct representations of $c$ as $F_k-p^\ell$ with $k\ge 2$ and $\ell\ge 0$. We prove that $m_{F,p}(c)\le 4$.