Diophantine equations concerning Terai's conjecture

This thesis consists of three chapters. In the first chapter, firstly some fundamental known notions from number theory, algebra and algebraic number theory are recalled. Then the notions such as the second order linear recurrence sequences, primitive divisor theorem and linear forms in logarithms which have and important place in the modern theory of Diophantine equations are given. In the second chapter of the thesis, it was shown that the Diophantine equation ((c+1)m^2+1)^x+(cm^2-1)^y=(am)^z under some conditions has only the positive integer solution (x,y,z)=(1,1,2). So, Terai's conjecture is confirmed for this equation. In the last chapter of the thesis, all positive integer solutions of the Diophantine equation (n-1)^x+(n+2)^y=n^z which is generalisation of Nagell's Diophantine equations 2^x+5^y=3^z and 4^x+7^y=5^z were found. The main tools which are used on the proofs are elementary methods of number theory, Baker's theory and primitive divisor theorem of Lucas sequences.