A parametric family of quartic Thue inequalities

Let c\neq 0,-1 be an integer. In this paper, we use the method of Tzanakis to transform the quartic Thue equation x^4 -(c^2+c+4) x^3y +(c^2+c+3) x^2 y^2 +2 xy^3 -y^4 = \mu into systems of Pell equations. Then, we determine all primitive solutions (x,y) with 0<|\mu|\leq |c+1|.

On Pillai’s problem with X-coordinates of Pell equations and powers of 2 II

In this paper, we show that if [Formula: see text] is the [Formula: see text]th solution of the Pell equation [Formula: see text] for some non-square [Formula: see text], then given any integer [Formula: see text], the equation [Formula: see text] has at most [Formula: see text] integer solutions [Formula: see text] with [Formula: see text] and [Formula: see text], except for the only pair [Formula: see text]. Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai’s problem in linear recurrent sequences.

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

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]