Sums of S-units in X-coordinates of Pell equations

TL;DR

This paper investigates the finiteness of solutions to Pell equations where multiple $X$-coordinates can be expressed as sums of $S$-units, providing explicit solutions for a specific case involving powers of two and three.

Contribution

It establishes the finiteness of certain Pell solutions with $S$-unit sum representations and explicitly solves a particular case involving powers of two and three.

Findings

Finiteness of $d$ with multiple $X$-coordinates as $S$-unit sums.

Explicit solution for the case with powers of two and three.

Characterization of solutions in terms of $S$-unit representations.

Abstract

Let $S$ be a fixed set of primes and let $(X_{l})_{l\geq 1}$ be the $X$-coordinates of the positive integer solutions $(X, Y)$ of the Pell equation $X^2-dY^2 = 1$ corresponding to a non-square integer $d>1$. We show that there are only a finite number of non-square integers $d>1$ such that there are at least two different elements of the sequence $(X_{l})_{l\geq 1}$ that can be represented as a sum of $S$-units with a fixed number of terms. Furthermore, we solve explicitly a particular case in which two of the $X$-coordinates are product of power of two and power of three.