On the solutions of some diophantine equations

In this work consisting of six chapters, we study one of the subjects of the Number Theory, Diophantine equations that has been popular for centuries. In the first section, the basic concepts, used later in the next chapters, are defined and some of the features related to these concepts are discussed. In the second part, general method for finding integer solutions of Diophantine equation x^2+C=y^n where C is an integer is examined. In the third chapter x^2+2^k=y^n Diophantine equation where k is odd, in the fourth chapter x^2+3^k=y^n Diophantine equation, fifth chapter x^2+5^k=y^n Diophantine equations, and finally in the sixth chapter Ramanujan-Nagell equation, also called x^2+7^k=2^n Diophantine integer solutions of the equation are sought. The work is a compilation.