Powers of two as sums of two $k-$Fibonacci numbers

TL;DR

This paper investigates which powers of two can be expressed as sums of two $k$-Fibonacci numbers, extending previous research using advanced number theory techniques.

Contribution

It extends prior work by applying linear forms in logarithms and Baker-Davenport reduction to identify powers of two as sums of two $k$-Fibonacci numbers.

Findings

Identifies specific powers of two that are sums of two $k$-Fibonacci numbers.

Provides bounds and conditions for such representations.

Extends previous classifications with new results.

Abstract

For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n}$ be the $k-$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 search for powers of 2 which are sums of two $k-$Fibonacci numbers. The main tools used in this work are lower bounds for linear forms in logarithms and a version of the Baker--Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of \cite{BL2} and \cite{BL13}.