Monatshefte für Mathematik

, Volume 185, Issue 1, pp 103–131

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

Article

## Abstract

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

### Keywords

Lucas sequences S-units Automatic resolution

### Mathematics Subject Classification

11D61 11B39 11Y50

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