Modular and reciprocity approaches to a family of diophantine equations

Abstract

In this thesis we study the Diophantine equation xp - Dy2p = z2; gcd(x; z) = 1; p prime: We combine two approaches: - The modular approach using in Wiles's proof of Fermat's Last Theorem. - Elementary quadratic reciprocity. We show how using this combination of approaches and computer calculations we can get congruence conditions for the exponent p