On the equation \(1+q+q^2+\cdots +q^{n-1}=p^m\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 0971.11513
author: Zhitang Guo
author: Yunfei Wu
publication date: 2001-09-20T00:00:00Z
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1649679
zbMATH Keywords: exponential Diophantine equations
MaRDI profile type: Publication
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title: On the equation \(1+q+q^2+\cdots +q^{n-1}=p^m\)
published in: Journal of Harbin Institute of Technology
review text: 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)].
reviewed by: Le Maohua
description: scientific article; zbMATH DE number 1649679