An old and new approach to Goormaghtigh’s equation

Bennett, Michael; Gherga, Adela; Kreso, Dijana

doi:10.1090/tran/8103

An old and new approach to Goormaghtigh’s equation
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by Michael A. Bennett, Adela Gherga and Dijana Kreso PDF

Trans. Amer. Math. Soc. 373 (2020), 5707-5745 Request permission

Abstract:

We show that if $n \geq 3$ is a fixed integer, then there exists an effectively computable constant $c (n)$ such that if $x$, $y$, and $m$ are integers satisfying \begin{equation*} \frac {x^m-1}{x-1} = \frac {y^n-1}{y-1}, \; \; y>x>1, \; m > n, \end{equation*} with $\gcd (m-1,n-1)>1$, then $\max \{ x, y, m \} < c (n)$. In case $n \in \{ 3, 4, 5 \}$, we solve the equation completely, subject to this non-coprimality condition. In case $n=5$, our resulting computations require a variety of innovations for solving Ramanujan-Nagell equations of the shape $f(x)=y^n$, where $f(x)$ is a given polynomial with integer coefficients (and degree at least two), and $y$ is a fixed integer.

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