On a problem of Pillai with $k$-generalised Fibonacci numbers and powers of $2$

TL;DR

This paper investigates the representations of integers as differences between k-generalized Fibonacci numbers and powers of 2, extending previous results for k=2 and k=3 to all k ≥ 4.

Contribution

It determines all integers with multiple representations as such differences for fixed k ≥ 4, generalizing earlier specific cases.

Findings

Identifies all integers with multiple representations for k ≥ 4

Extends previous results from k=2 and k=3 to higher k

Provides a complete characterization of these differences

Abstract

For an integer $ k\geq 2 $, let $ \{F^{(k)}_{n} \}_{n\geq 0}$ be the $ k$--generalized Fibonacci sequence which starts with $ 0, \ldots, 0, 1 $ ($ k $ terms) and each term afterwards is the sum of the $ k $ preceding terms. In this paper, we find all integers $c$ having at least two representations as a difference between a $k$--generalized Fibonacci number and a powers of 2 for any fixed $k \geqslant 4$. This paper extends previous work from [9] for the case $k=2$ and [6] for the case $k=3$.