Generalized Goldbachs Conjectures and the Arithmetic of Elliptic Curves

Abstract

In this thesis, we introduce a totally new method which constructs integral points of elliptic curves. The main idea is that results on the theory of excep- tional sets of linear type generalizations of strong Goldbach conjecture show the existence of integral points of certain elliptic curves, which are called cubic twists. We first show that the size of exceptional sets of linear type generaliza- tions of strong Goldbach conjecture is relatively small by generalizing previous works. We also explain why this result can be used for constructing integral points of elliptic curves. After that we give two applications about the theory of elliptic curves. One is that there are infinitely many sums of two rational cubes which have arbitrary number of prime divisors. The other is that under the parity conjecture, there are infinitely many elliptic curves whose Mordell–Weil groups are exactly Z × Z.