Closed formulae for certain Fermat-Pell equations

2021/07/06 by Szechtman, Fernando
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2107.02696

Abstract

Given positive integers j,k, with j≥ 2, we show that there are positive integers d,e such that √(d) has continued fraction expansion √(d)=[e,k,…,k,2e], with period j, if and only if k is even or 3\nmid j, in which case we give closed formulae to find all such d,e as well as the smallest solution in positive integers to the Fermat-Pell equation X²-dY²=(-1)^j.

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