Linear combinations of prime powers in sums of terms of binary recurrence sequences

TL;DR

This paper proves finiteness results for solutions to certain Diophantine equations involving sums of binary recurrence sequence terms equaling sums of prime powers, and explicitly solves a specific case with powers of 2 and 3.

Contribution

It establishes a general finiteness theorem for these equations and explicitly solves a particular instance involving Fibonacci numbers and prime powers.

Findings

Finiteness of solutions for the general Diophantine equation under certain conditions

Explicit solution to the specific equation involving Fibonacci numbers and prime powers

Application of linear forms in logarithms and Baker-Davenport method

Abstract

Let $\{ {U_{n}\}_{n \geq 0} }$ be a non-degenerate binary recurrence sequence with positive discriminant. Let $\{p_1,\ldots, p_s\}$ be fixed prime numbers and $\{b_1,\ldots ,b_s\}$ be fixed non-negative integers. In this paper, we obtain the finiteness result for the solution of the Diophantine equation $U_{n_{1}} + \cdots + U_{n_{t}} = b_1 p_1^{z_1} + \cdots+ b_s p_s^{z_s} $ under certain assumptions. Moreover, we explicitly solve the equation $F_{n_1}+ F_{n_2}= 2^{z_1} +3^{z_2}$, in non-negative integers $n_1, n_2, z_1, z_2$ with $z_2\geq z_1$. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker-Davenport reduction method.