Effective Resolution of Diophantine equations of the form $u_n+u_m=w p_1^{z_1} \cdots p_s^{z_s}$

TL;DR

This paper establishes effective finiteness results and explicit bounds for solutions to a class of Diophantine equations involving binary recurrence sequences and prime power products, along with an efficient solving algorithm.

Contribution

It provides the first effective bounds and an algorithm for solving equations of the form u_n + u_m = w p_1^{z_1} ... p_s^{z_s} with fixed recurrence and primes.

Findings

Explicit upper bounds for n, m, and z_i are derived.

An efficient algorithm for solving these equations is developed.

The method is demonstrated through a concrete example.

Abstract

Let $u_n$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_1,p_2,\dots,p_s$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $u_n+u_m=w p_1^{z_1} \cdots p_s^{z_s}$ and prove under mild technical restrictions effective finiteness results. In particular we give explicit upper bounds for $n,m$ and $z_1, \dots, z_s$. Furthermore, we provide a rather efficient algorithm to solve Diophantine equations of the described type and we demonstrate our method by an example.