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A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING PRIMITIVE PYTHAGOREAN TRIPLES

Part of: Diophantine equations

Published online by Cambridge University Press:  26 September 2016

MOU-JIE DENG  and
DONG-MING HUANG
MOU-JIE DENG*
Affiliation:
Department of Applied Mathematics, Hainan University, Haikon, Hainan 570228, PR China email Moujie_Deng@163.com
DONG-MING HUANG
Affiliation:
Department of Applied Mathematics, Hainan University, Haikon, Hainan 570228, PR China email Huangdm35@126.com
*
✉Moujie_Deng@163.com
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Abstract

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Let $a,b,c$ be a primitive Pythagorean triple and set $a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$, where $m$ and $n$ are positive integers with $m>n$, $\text{gcd}(m,n)=1$ and $m\not \equiv n~(\text{mod}~2)$. In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation $(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ is $(x,y,z)=(2,2,2)$. We use biquadratic character theory to investigate the case with $(m,n)\equiv (2,3)~(\text{mod}~4)$. We show that Jeśmanowicz’ conjecture is true in this case if $m+n\not \equiv 1~(\text{mod}~16)$ or $y>1$. Finally, using these results together with Laurent’s refinement of Baker’s theorem, we show that Jeśmanowicz’ conjecture is true if $(m,n)\equiv (2,3)~(\text{mod}~4)$ and $n<100$.

Keywords

exponential Diophantine equationsPythagorean triplesJeśmanowicz’ conjecture

MSC classification

Primary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 1 , February 2017 , pp. 5 - 13
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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