| In this thesis we have studied the following group of quartic Diophantine equations. x4-kx2y2 +y4=2j where k∈N,j∈&cubl0;1 ,2,3,4,5,6,7,8&cubr0;; These equations are examples of Thue-Mahler equations. Theory of p-adic linear forms can be used to determine the solutions for such equations when k > 2. This approach is computationally intensive and a bit restrictive as we can deal with only one value of k at a time.; In the present work we have used the solutions for various Pell's equations (x2 - Dy2 = N) and the bounds on their fundamental solutions in each class to show that the above group of equations cannot have any positive integer solution except when k = 2 or when k is a square on an integer.; We conclude the thesis by a conjecture that there is no positive integer solution to the equation x4 - kx2y2 + y 4 = 2j for all j ∈N . We believe that with further knowledge of solutions of special types of Pell's equations (x2 - Dy 2 = 2n) a descent method can be developed to solve all the cases. |
