Rocky Mountain Journal of Mathematics

Upper bounds for solutions of an exponential Diophantine equation

Takafumi Miyazaki

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We consider the exponential Diophantine equation $a^{x}+b^{y}=c^{z}$ in positive integers $x$, $y$ and $z$, where $a$, $b$ and $c$ are fixed pair-wise relatively prime positive integers greater than one. In this paper, we obtain several upper bounds for solutions $x$, $y$ and $z$ for which two of $x$, $y$ and $z$ are even. As their applications, we solve exponential Diophantine equations in which $a$, $b$ and $c$ are expressed as terms of linearly recurrence sequences.

Article information

Rocky Mountain J. Math., Volume 45, Number 1 (2015), 303-344.

First available in Project Euclid: 7 April 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11D61: Exponential equations
Secondary: 11D41: Higher degree equations; Fermat's equation

Exponential Diophantine equations generalized Fermat equations $abc$-conjecture


Miyazaki, Takafumi. Upper bounds for solutions of an exponential Diophantine equation. Rocky Mountain J. Math. 45 (2015), no. 1, 303--344. doi:10.1216/RMJ-2015-45-1-303.

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