方程(ax^m+1)/(ax+1)=y^n的正整数解

Positive Integer Solutions of the Equation (ax^m+1)/(ax+1)=y^n

设a,m是大于1的正整数,证明:当m>2,方程(axm+1)/(ax+1)=yn仅有有限多组解(x,y,n)适合min(x,y,n)>1,而且这些解都满足yn<xm-1≤am2-3m+2.

Let a and m be positive integers with min（a, m）〉 1. In this paper we prove that if m〉2, then the equation （ax^m＋1）/（ax＋l）=y^n has only finitely many positive integer solution （x, y, n） with min（x, y, n） 〉 1. Moreover, all the solutions satisfy y^n 〈 x^m-1 ≤ a^m2-3m＋2.