Tokyo Journal of Mathematics

A Note on the Shuffle Variant of Jeśmanowicz' Conjecture

Zsolt RÁBAI

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Abstract

Let $(a,b,c)$ be a primitive Pythagorean triple. In 1956, Jeśmanowicz conjectured that the equation $a^{x}+b^{y}=c^{z}$ has the unique solution $(x,y,z)=(2,2,2)$ in positive integers. In 2010 Miyazaki proposed a similar problem. He conjectured that if $(a,b,c)$ is again a primitive Pythagorean triple with $b$ even, then the equation $c^{x}+b^{y}=a^{z}$ with $x$, $y$ and $z$ positive integers has the unique solution $(x,y,z)=(1,1,2)$ if $c=b+1$ and no solutions if $c>b+1$. He also proved that his conjecture is true if $c \equiv 1 \pmod{b}$. We extend Miyazaki's result to the case $c \equiv 1 \pmod{b/2^{\textrm{ord}_{2}(b)}}$.

Article information

Source
Tokyo J. Math., Volume 40, Number 1 (2017), 153-163.

Dates
First available in Project Euclid: 8 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1502179220

Digital Object Identifier
doi:10.3836/tjm/1502179220

Mathematical Reviews number (MathSciNet)
MR3689983

Zentralblatt MATH identifier
06787092

Subjects
Primary: 11D61: Exponential equations

Citation

RÁBAI, Zsolt. A Note on the Shuffle Variant of Jeśmanowicz' Conjecture. Tokyo J. Math. 40 (2017), no. 1, 153--163. doi:10.3836/tjm/1502179220. https://projecteuclid.org/euclid.tjm/1502179220


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