Han Di and Guan Wenji: A note on the Diophantine equation $(x^{p}-1)/(x-1)=p^{e}y^{q}$, p.35-43


Let $p, q$ be odd primes, and let $e\in\{0,
1\}$. In this paper, using a lower bound for two logarithms in the complex case, we prove that if $p\equiv 3\ (\bmod 4)$ and $q>220p(\log p)^{2}$, then the equation $(x^{p}-1)/(x-1)=p^{e}y^{q}$ has no positive integer solution $(x,
y)$ with $\min\{x, y\}>1$.

Key Words: Higher diophantine equation, Nagell-Ljunggren equation, Gel$^{'}$fond-Baker method.

2000 Mathematics Subject Classification: Primary: 11D41
Secondary: 11D45.

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