In 1993, the reviewer [Acta Arith. 64, 19-28 (1993; Zbl 0783.11013)] proved that the equation \((*)\) \((x^m-1)/(x-1)= y^n\), \(x,y\in \mathbb{N}\), \(x>1\), \(y>1\), \(m>2\), \(n>1\), has only finitely many solutions \((x,y,m,n)\) with \(x\) a prime power and \(y\equiv 1\pmod x\). It implies that \((*)\) has only finitely many solutions \((x,y,m,n)\) satisfying \((**)\) \(x= p^l\) and \(\text{gcd} (n,p-1)= 1\), where \(p\) is a prime and \(l\) is a positive integer.
In the present paper, the author gives some effectively computable upper bounds. He proves that if \((x,y,m,n)\) is a solution of \((*)\) satisfying \((**)\), then \(n\leq 1900\). Further, if \(p\nmid n\), then \(x\leq 4063\).