Borka Jadrijević

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 . 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 and solve it by the method of Tzanakis under the same assumptions.

Solving a family of quartic Thue inequalities using continued fractions

In this paper we find all primitive solutions of the Thue inequality |x4 + 2(1− n)xy + y4| ≤ 2n + 3, where n ≥ 0 is an integer.

DETERMINING ELEMENTS OF MINIMAL INDEX IN AN INFINITE FAMILY OF TOTALLY REAL BICYCLIC BIQUADRATIC NUMBER FIELDS

Let c ≠ 2 be a positive integer such that c and c + 4 are square-free. We consider the infinite parametric family of bicyclic biquadratic fields K = Q(sqrt {; ; 2c}; ; , sqrt {; ; 2(c + 4)}; ; ). We determine the integral basis of the field. We show that K admits no power integral basis, determine the minimal index and all elements of minimal index. We use the solutions of a parametric family of quartic Thue equations and extensive numerical calculations by Maple and Magma are also involved.