On the equation \(1+q+q^2+\cdots +q^{n-1}=p^m\) (Q2742041)

Let \(p\), \(q\) be odd primes, and let \(D=pq(q-1)\). In this paper, by some elementary methods, the authors prove mainly that the equation of the title has integer solutions \((m,n)\) with \(n\geq 5\), \(m\geq 2\), \(2\nmid m\) if and only if \(2\nmid n\) and the fundamental solution \(\epsilon\) of the equation \(u^2-Dv^2=1\) satisfies \(\epsilon =q^n+p^m(q-1)+2p^{(m-1)/2} q^{(n-1)/2} \sqrt D\). The above result is Exercise 2.6.5 in a book of \textit{Z. Cao} [Introduction to Diophantine equations (Chinese), Haerbin Gongye Daxue Chubanshe, Harbin (1989; Zbl 0849.11029)].