关于Diophantine方程4x^(2n)-py^2=1

On the Diophantine equation 4x^(2n)-py^2=1

设n>1是正整数,p是大于3的奇素数.本文运用初等数论的方法,结合广义Lebesgue-Nagell方程和广义Fermat方程的性质,研究了丢番图方程4x2n-py2=1的整数解,并证明了对于任意奇数n,此方程没有正整数解(x,y).

Let n be a positive integer with n〉1,and let pbe an odd prime with p〉3.By using the elementary and the properties of the generalized Lebesgue-Nagell equations and the generalized Fermat equations,the integer solutions of the equation 4x2n-py2=1is studied.It is prove that this equation has no positive integer solution（x,y）for 2n.