关于指数Diophantine方程x^3-1=2py^2

On the Diophantine equation x^3-1=2py^2

设p是6k+1型的奇素数,运用Pell方程px2-3y2=1的最小解、同余式、平方剩余、勒让德符号的性质等初等方法证明了当p=3n(n+1)+1≡1,7(mod8)(n为单数)为奇素数,且2n+1为奇素数时,指数Diophantine方程x3-1=2py2无正整数解.

Let p be an odd prime of the form 6k＋1. Using the elementary method of the smallest solution of the Pell equation px2-3y2=1, congruent formula, quadratic residue, and the nature of legendre sign,this paper obtains a sufficient condition that the Diophantine equation x3- 1≡2py2 has no integer solutions, where p=3n（n＋ 1）＋ 1 ≡ 1, 7（mod8）, n is a positive odd integer, P and 2n＋ 1 are odd prime.