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A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING NONPRIMITIVE PYTHAGOREAN TRIPLES

Part of: Diophantine equations Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  21 October 2020

YASUTSUGU FUJITA Open the ORCID record for YASUTSUGU FUJITA [Opens in a new window]  and
MAOHUA LE Open the ORCID record for MAOHUA LE [Opens in a new window]
YASUTSUGU FUJITA*
Affiliation:
Department of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, Japan
MAOHUA LE
Affiliation:
Institute of Mathematics, Lingnan Normal College, Zhangjiang, Guangdong, 524048, China e-mail: lemaohua2008@163.com
*
e-mail: fujita.yasutsugu@nihon-u.ac.jp
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Abstract

Jeśmanowicz conjectured that $(x,y,z)=(2,2,2)$ is the only positive integer solution of the equation $(*)\; ((\kern1.5pt f^2-g^2)n)^x+(2fgn)^y=((\kern1.5pt f^2+g^2)n)^x$ , where n is a positive integer and f, g are positive integers such that $f>g$ , $\gcd (\kern1.5pt f,g)=1$ and $f \not \equiv g\pmod 2$ . Using Baker’s method, we prove that: (i) if $n>1$ , $f \ge 98$ and $g=1$ , then $(*)$ has no positive integer solutions $(x,y,z)$ with $x>z>y$ ; and (ii) if $n>1$ , $f=2^rs^2$ and $g=1$ , where r, s are positive integers satisfying $(**)\; 2 \nmid s$ and $s<2^{r/2}$ , then $(*)$ has no positive integer solutions $(x,y,z)$ with $y>z>x$ . Thus, Jeśmanowicz’ conjecture is true if $f=2^rs^2$ and $g=1$ , where r, s are positive integers satisfying $(**)$ .

Keywords

Jeśmanowicz’ conjecturePythagorean tripleternary purely exponential Diophantine equation

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11J86: Linear forms in logarithms; Baker's method
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 29 - 39
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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