# ON THE DIOPHANTINE EQUATIONS n^x-n^y=z^2 AND 2^x-p^y=z^2

## Authors

• Napalai Laomalaw

## Keywords:

Diophantine equation, Integer solution, Catalan’s conjecture

## Abstract

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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