Some quartic diophantine equations
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In this thesis we have studied the following group of quartic Diophantine equations. [Special characters omitted.] <display-math> <fd> <fl>x<sup><rm>4</rm></sup>-kx<sup><rm>2</rm></sup>y<sup><rm>2 </rm></sup>+y<sup><rm>4</rm></sup>=<rm>2<sup><mit>j</mit></sup> <hsp sp="1.000"><hsp sp="0.212">where<hsp sp="1.000"><hsp sp="0.212"> <mit>k&isin;<blkbd>N,<hsp sp="1.000"><mit>j&isin;&cubl0;<rm>1 ,2,3,4,5,6,7,8&cubr0;</rm></mit></blkbd></mit></rm></fl> </fd> </display-math> 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 ( x 2 - Dy 2 = 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 x 4 - kx 2 y 2 + y 4 = 2 j for all j [Special characters omitted.] <math> <f> &isin;<sc><blkbd>N</blkbd></sc></f> </math> . We believe that with further knowledge of solutions of special types of Pell's equations ( x 2 - Dy 2 = 2 n ) a descent method can be developed to solve all the cases.