Let \(x,m,n\in \mathbb{N}\) be such that \(x>1\) and \(n>1\), and let \(u_ m(x)= x^{m-1}+ \cdots+ x+1\). T. N. Shorey [Math. Proc. Camb. Philos. Soc. 99, 195-207 (1986; Zbl 0598.10029)] proved that if \(m>1\), \(m\equiv 1\pmod n\) and \(u_ m (x)\) is an \(n\)-th power, then \(\max (x,m,n)< C\), where \(C\) is an effectively computable absolute constant. This problem relates to the solutions \((x,y, m,n)\) of the equation \[ (x^ m- 1)/( x- 1)= y^ n, \qquad x,y, m,n\in \mathbb{N}, \quad x>1, \quad y>1, \quad m>2, \quad n>1. \tag \(*\) \] In this paper, the author proves that \((*)\) has no solution \((x,y, m,n)\) satisfying \(\text{gcd} (x\varphi (x), n)=1\), where \(\varphi(x)\) is Euler’s function of \(x\). As a consequence, the author shows that if \(m>1\), \(m\equiv 1\pmod n\) and \(u_ m (x)\) is an \(n\)-th power, then \((x,m,n)= (3,5,2)\). The proof uses some algebraic number theory and diophantine approximation methods.
The main result in this paper has contributed to solving many problems concerning the equation \((*)\). For instance, in a forthcoming paper [Math. Proc. Camb. Philos. Soc. 116, No. 3, 385-389 (1994)], the author proves that all solutions \((x,y, m,n)\) of \((*)\) satisfy \(n= \text{ord}_ y x\). This solves a problem by H. Edgar [Rocky Mt. J. Math. 15, 327-329 (1985; Zbl 0583.10012)].