关于Diophantine方程(ax^m-1)/(ax-1)=y^n

On diophantine equation (ax^m-1)/(ax-1)=y^n

设a,m,n是适合m in(m,n)>2的正整数。证明了:当m≡1(m od n)时,方程(axm-1)/(ax-1)=yn无正整数解(x,y)适合m in(x,y)>1。

Let a,m,n be positive integers with min(m,n)m>2.In this paper we prove that if m≡1(mod n),then the equation (ax^m-1)/(ax-1)=y^n has no positive integer solution (x,y) with min(x,y)>1.