## Functiones et Approximatio Commentarii Mathematici

### Jeśmanowicz' conjecture on exponential diophantine equations

Takafumi Miyazaki

#### Abstract

Let $(a,b,c)$ be a primitive Pythagorean triple such that $a^2+b^2=c^2$ with even $b$. In 1956, L. Jeśmanowicz conjectured that the equation $a^x+b^y=c^z$ has only the solution $(x,y,z)=(2,2,2)$ in positive integers. In this paper, we give various new results on this conjecture. In particular, we prove that if the equation has a solution $(x,y,z)$ with even $x,z$ then $x/2$ and $z/2$ are odd.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 45, Number 2 (2011), 207-229.

Dates
First available in Project Euclid: 12 December 2011

https://projecteuclid.org/euclid.facm/1323705814

Digital Object Identifier
doi:10.7169/facm/1323705814

Mathematical Reviews number (MathSciNet)
MR2895155

Zentralblatt MATH identifier
1266.11064

#### Citation

Miyazaki, Takafumi. Jeśmanowicz' conjecture on exponential diophantine equations. Funct. Approx. Comment. Math. 45 (2011), no. 2, 207--229. doi:10.7169/facm/1323705814. https://projecteuclid.org/euclid.facm/1323705814

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