Prime powers in sums of terms of binary recurrence sequences

TL;DR

This paper investigates the finiteness of solutions to sums of binary recurrence sequence terms equaling prime powers, explicitly finds all powers of three as sums of three balancing numbers, and applies advanced Diophantine approximation techniques.

Contribution

It establishes finiteness results for certain Diophantine equations involving binary recurrence sequences and prime powers, and explicitly characterizes solutions for specific cases.

Findings

Finiteness of solutions for the sum of recurrence sequence terms equaling prime powers.

Explicit solutions for powers of three as sums of three balancing numbers.

Application of Baker-Davenport reduction method in solving these equations.

Abstract

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine equation $u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}$ with some conditions on $n_i $ for all $1\leq i \leq t$. Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using the lower bounds for linear forms in logarithms. Further, we use a variant of Baker-Davenport reduction method in Diophantine approximation due to Dujella and Peth\H{o}.