On X -Coordinates of Pell Equations that Are Repdigits

2018/02/01 by Bernadette Faye, Florian Luca
Mathematics · Computer Science · Physics and Astronomy · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Nonlinear Waves and Solitons

paper · doi:10.1080/00150517.2018.12427721

Abstract

Let b ≥ 2 be a given integer. In this paper, we show that there are only finitely many positive integers d that are not squares, such that the Pell equation X2 − dY2 = 1 has two positive integer solutions (X, Y) with the property that their X-coordinates are base b-repdigits. Recall that a base b-repdigit is a positive integer whose digits have the same value when written in base b. We also give an upper bound on the largest such d in terms of b.

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