ON THE DIOPHANTINE EQUATIONS n^x-n^y=z^2 AND 2^x-p^y=z^2
Keywords:
Diophantine equation, Integer solution, Catalan’s conjectureAbstract
In this paper, we study two Diophantine equations and where is a positive integer with and is a prime number in order to generate all non-negative integer solutions . This can be shown as follows: the Diophantine equation has all solutions in the following form, where is a non-negative integer. For the Diophantine equation, we show that 1) if , then this equation has all solutions in the form , where is a non-negative integer and is an integer, 2) if , then this equation has only three solutions and 3) if and or, then this equation has only two solutions
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