Solutions Of Pell Equations Research Articles

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]

On special extrema of polynomials with applications to Diophantine problems

We prove that apart from explicitly given cases, described in terms of Dickson polynomials, a polynomial fin mathbb {Q}[x] can have at most one shift f(x)-lambda (lambda in mathbb {C}) of the form u(g(x))^q(h(x))^k with uin mathbb {C}, g,hin mathbb {C}[x] and either deg (g)=2, k is even, q=k/2 or deg (g)le 1, kge 2, qge 1. This is shown by handling the case of two possible shifts, which was an open issue. As an application, we give a precise statement yielding a description of polynomials f having infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely many S-integral solutions when the polynomial involved is composite. In the case where there are finitely many solutions, our results yield effective bounds for them. Finally, as further applications, we give effective results for polynomial values in the solutions of Pell equations and in non-degenerate binary recurrence sequences.

English

English

On periodic continued fractions, Pell equation, and Fermat's challenge numbers

A method for direct construction of continued fraction elements, presenting a square root from a natural number, is proposed. This technique resembles the usual division process. Derivation of the computing procedures is based on the Euclidean algorithm, Fermat's and Euler's theorems on a sum of two squares, and formulation of the basic equation for the continued fraction elements as Diophantine equation. The solution of Pell equation is presented explicitly through the same functions, which describe the Euclidean algorithm and solutions to linear Diophantine equations. It is supposed that similar procedures could be used by Fermat when he challenged to solve the quadratic Diophantine equation for special difficult cases found by him.

On the least solution of Pell’s equation

On the least solution of Pell’s equation