Some problems in number theory [HBNI Th50]

Abstract

In this thesis the author has worked on three different problems. Some progress is reported on these three problems. The first problem considered is about "Higher Residue Symbols". Given a finite set S of integers, the question of finding primes p such that each integer, 'a (an element of) S' is a quadratic residue (non-residue) modulo p is dealt by various authors. Many authors including M. Fried and S. Wright have established the infinitude of primes p modulo which each "a an element of S" is a quadratic residue. The density of such primes was considered in for study. The author has generalized the problem and studied the analogous questions. The second problem considered is the Catalan's conjecture / Mihailescu's Theorem. It was conjectured by Eugene Charles Catalan in 1844 that, the only perfect powers among integers which differ by 1 are 8 and 9. As part of this thesis the author studies the equation (x)^p - (y)^q = 1 over a number field K, i.e. when x, y Elements of (O)k. Theory of 'torsion points on elliptic curves' is used to handle the equation when one of the prime is even. The chapter four formulates an appropriate Cassels criteria and prove it partially for imaginary quadratic number fields with class number one. Chapter five reports further progress made on Catalan problem considered here. The author introduces a proper obstruction group, made up of solutions of Catalan problem, and then trap it in a short exact sequence of fairly well studied objects (namely class groups and unit groups). This is pretty analogous to the work in the case of Catalan's conjecture over Z.