On some exponential diophantine equations

In this three-part study, a special case of Diophantine equations, which is one of the problems of the number theory as old as human history, is discussed. More precisely, for the exponential Diophantine equations, Jeśmanowicz's Conjecture, which is an open problem in the literature, is studied. Sierpinski proved that the exponential Diophantine equation (2, 2, 2) is 3^x+4^y=5^z by looking at in a different perspective to the 2500-years old Pythagorean Theorem. In the same year, Jeśmanowicz claimed that there was only one solution (2, 2, 2) of the exponential Diophantine equation, a^x+b^y=c^z with a^2+b^2=c^2. This assumption has been confirmed in many cases and the exact solution is not yet known. In the first part, exponential Diophantine equations are introduced and in the second part, this assumption is studied. A large-scale survey was carried out in the literature. In the third chapter, it is seen that the only solution (2, 2, 2) of the exponential Diophantine equation (20k)^x+(99k)^y=(101k)^z, which is a special case of Jesmanowicz's conjecture by elementer methods. The study is compilation.