On X-coordinates of Pell equations which are repdigits

2016/05/12 by Faye, Bernadette, Luca, Florian
#11D45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1605.03756

Abstract

Let b≥ 2 be a given integer. In this paper, we show that there only finitely many positive integers d which 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 all 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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