Functiones et Approximatio Commentarii Mathematici

Jeśmanowicz' conjecture on exponential diophantine equations

Takafumi Miyazaki

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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

Permanent link to this document
https://projecteuclid.org/euclid.facm/1323705814

Digital Object Identifier
doi:10.7169/facm/1323705814

Mathematical Reviews number (MathSciNet)
MR2895155

Zentralblatt MATH identifier
1266.11064

Subjects
Primary: 11D61: Exponential equations
Secondary: 11D41: Higher degree equations; Fermat's equation 11J86: Linear forms in logarithms; Baker's method

Keywords
exponential diophantine equations lower bounds for linear forms in logarithms of algebraic numbers generalized Fermat equations Pythagorean triples

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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