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

2024/11/17 by Nair, Parvathi S, Rout, Sudhansu Sekhar
#11B37 (Primary) 11D45 #11J86 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.11103

Abstract

Let S be a fixed set of primes and let (X_l)_{l≥ 1} be the X-coordinates of the positive integer solutions (X, Y) of the Pell equation X²-dY² = 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≥ 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.

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