New Conditions on the Nagell-Ljunggren Equation

\frac{x^p-1}{x-1}=y^q

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Abstract

In this paper, the authors consider the equation

\frac{x^p-1}{x-1}=p^e{y}^q

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{{x^p} - 1} \over {x - 1}} ={p^e}{y^q}$$\end{document}

, for distinct odd prime exponents p, q, and show that, for p > 3, it has no solutions under the condition that q does not divide h_p⁻, the minus part of the class number of the p-th cyclotomic field.

Keywords

Nagell-Ljunggren conjecture / Diophantine equations / Class field theory / 11D61 / 11R37

References

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