A note on the exponential diophantine equation $a^x + b^y = c^z$

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A note on the exponential diophantine equation $a^x + b^y = c^z$

Maohua Le

Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 80, Number 4 (2004), 21-23.

Abstract

Let $a$, $b$, $c$ be fixed coprime positive integers. In this paper we prove that if $b \equiv 3 \pmod{4}$, $a \equiv -1 \pmod{b^{2l}}$, $a^2 + b^{2l-1} = c$ and $c$ is odd, where $l$ is a positive integer, then the equation $a^x + b^y = c^z$ has only the positive integer solution $(x,y,z) = (2,2l-1,1)$.

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Primary Subjects: 11D61

Keywords: Exponential diophantine equations; primitive divisors of Lucas numbers

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.pja/1116442210
Mathematical Reviews number (MathSciNet): MR2055070
Zentralblatt MATH identifier: 1050.11040
Digital Object Identifier: doi:10.3792/pjaa.80.21

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