**Thesis Type:** Doctorate

**Institution Of The Thesis:** Uludağ Üniversitesi, Turkey

**Approval Date:** 2007

**Thesis Language:** Turkish

**Student:** MUSA DEMİRCİ

**Supervisor: **İSMAİL NACİ CANGÜL

In this thesis, two special classes of two variable cubic Diophantine equations, called Bachet and Frey equations, are considered in relation with some elliptic curve classes. A new set of results related to the solutions of Diophantine equations corresponding to the ones obtained for elliptic curve classes is given. The number of rational points on Bachet elliptic curves y2=x3+a3, and Frey elliptic curves y2=x3-n2x, which are just some special cases of simplified Weierstrass equation; their orders, and the group structure of them are considered. As the rational points on elliptic curves correspond to the solutions of the Diophantine equations, the results obtained for elliptic curves are also valid for the corresponding Diophantine equations. In the first two chapters, the preliminary information necessary for the second and third chapters are recalled. The notions of Diophantine equations and elliptic curves are defined and the relations between them are obtained. In the second chapter, some results concerning the number of rational points on Bachet and Frey elliptic curves are given. In the third chapter, the group structure of the solution sets of these Diophantine equations under the addition operation are considered.