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Jeśmanowicz' conjecture and related equations

Volume 184 / 2018

Pingzhi Yuan, Qing Han Acta Arithmetica 184 (2018), 37-49 MSC: Primary 11D61; Secondary 11D41. DOI: 10.4064/aa170508-17-9 Published online: 12 April 2018

Abstract

Jeśmanowicz conjectured that the exponential Diophantine equation $(m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z$ has only the positive integer solution $(x,y, z)=(2,2,2)$, where $m$ and $n$ are positive integers with $m \gt n$, $\gcd(m, n)=1$ and $m\not\equiv n\pmod{2}$. In this paper, we first improve the result of Miyazaki and Terai (2015). Let $a, b$ and $c$ be coprime integers with $a+b=c$. We also obtain some results on the exponential equation $(an)^x+(bn)^y=(cn)^z$.

Authors

  • Pingzhi YuanSchool of Mathematics
    South China Normal University
    Guangzhou 510631, China
    e-mail
  • Qing HanFaculty of Common Courses
    South China Business College
    of Guangdong University of Foreign Studies
    Guangzhou 510545, China
    e-mail

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