On the solvability of the simultaneous Pell equations x2 − ay2 = 1 and y2 − bz2 = v12

Abstract

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

On the cubic Pell equation over finite fields

“…For instance, the study of the size of the fundamental solution is an interesting problem addressed in several papers, e.g., [7,11,22]. Recently, the solvability of simultaneous Pell equations and explicit formulas for their solutions have been also studied in [14,8,12]. Moreover, it is also interesting to study the Pell equation over finite fields, determining the number of solutions and their properties [18,20,19,9].…”