On the Diophantine equation $x\sp {2n}-\mathcal{D}y^2=1$

Abstract

In this paper, it has been proved that if 2$"> and Pell's equation <!-- MATH ${u^2} - \mathcal{D}{v^2} = - 1$ --> has integer solution, then the equation <!-- MATH ${x^{2n}} - \mathcal{D}{y^2} = 1$ --> has only solution in positive integers , (when , <!-- MATH $\mathcal{D} = 122$ --> ). That is proved by studying the equations <!-- MATH ${x^p} + 1 = 2{y^2}$ --> and <!-- MATH ${x^p} - 1 = 2{y^2}$ --> ( is an odd prime). In addition, some applications of the above result are given.