Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Maohua Le

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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 $(x,y,z) = (2,2l-1,1)$.

Article information

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

Dates
First available in Project Euclid: 18 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1116442210

Digital Object Identifier
doi:10.3792/pjaa.80.21

Mathematical Reviews number (MathSciNet)
MR2055070

Zentralblatt MATH identifier
1050.11040

Subjects
Primary: 11D61: Exponential equations

Keywords
Exponential diophantine equations primitive divisors of Lucas numbers

Citation

Le, Maohua. A note on the exponential diophantine equation $a^x + b^y = c^z$. Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 4, 21--23. doi:10.3792/pjaa.80.21. https://projecteuclid.org/euclid.pja/1116442210


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References

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