Pellian equations of special type

In this paper, we consider the solvability of the Pellian equation x 2 − ( d 2 + 1 ) y 2 = − m , $$\begin{array}{} \displaystyle x^2-(d^2+1)y^2 = -m, \end{array} $$ in cases d = nk , m = n 2l−1, where k, l are positive integers, n is a composite positive integer and d = pq, m = pq 2, p, q are primes. We use the obtained results to prove results on the extendibility of some D(−1)-pairs to quadruples in the ring Z [ − t ] $\begin{array}{} {\mathbb{Z}}[\sqrt{-t}] \end{array} $ , with t > 0.

A SYSTEM OF RELATIVE PELLIAN EQUATIONS AND A RELATED FAMILY OF RELATIVE THUE EQUATIONS

In this paper we consider the family of systems (2c + 1)U2 - 2cV2 = μ and (c - 2)U2 - cZ2 = -2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field [Formula: see text]. We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation [Formula: see text] and solve it by the method of Tzanakis under the same assumptions.

On the extendibility of the Diophantine triple{1,5,c}

We study the problem of extendibility of the triples of the form{1,5,c}. We prove that ifck=sk2+1, where(sk)is a binary recursive sequence,kis a positive integer, and the statement that all solutions of a system of simultaneous Pellian equationsz2−ckx2=ck−1,5z2−cky2=ck−5are given by(x,y,z)=(0,±2,±sk), is valid for2≤k≤31, then it is valid for all positive integerk.

Simultaneous Pellian equations

In this paper we describe a method for finding integer solutions of simultaneous Pellian equations, that is, integer triples (x,y,z) satisfying equations of the formwhere the coefficientsa,b,c,d,fare integers and we assume thata,c, andacare not square.