广义Brocard-Ramanujan方程x^2-D=y!解的上界

The upper bound for solutions of the generalized Brocard-Ramanujan equation x^2-D=y!

设D是无平方因子正整数,即D为不含任意素数p的方幂.运用初等方法及二次非剩余的性质,讨论了广义Brocard-Ramanujan方程x2-D=y!的正整数解(x,y)的上界估计问题,证明了该方程的正整数解(x,y)都满足y<4槡DlogD.

Let D be a positive integer with square-free .For any prime p ,the square of p cloes not divide D .In this paper ,using the elementary method and the properties of quadratic non-residues ,the upper bound for positive integer solutions (x ,y) of the equation x2 -D= y!is discussed .It is proved that all the positive integer solutions (x ,y) of the equation satisfy y<4 Dlog D .