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 数学进展 - 2018, Vol. 47(3): 388-392
 研究论文
 关于丢番图方程 $(na)^x+(nb)^y=(nc)^z$ On the Diophantine Equation $(na)^x+(nb)^y=(nc)^z$ 陈凤娟 CHEN Fengjuan 苏州大学数学科学学院, 苏州, 江苏, 215006 School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, 215006, P. R. China 收稿日期: 2016-11-09 出版日期: 2018-06-08 2018, Vol. 47(3): 388-392 DOI: 10.11845/sxjz.2016129b
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Abstract：Let $(a,b,c)$ be a primitive Pythagorean triple such that $a^2+b^2=c^2$ with $2\,|\,b$. In 1956, Jesmanowicz conjectured that, for any positive integer $n$, the Diophantine equation $(na)^x+(nb)^y=(nc)^z$ has no positive solution $(x,y,z)$ other than $x=y=z= 2$. Denote by $P(n)$ the product of distinct prime factors of $n$. In this paper, we prove that the conjecture is true for $(a, b, c)=( p^{2r}-4, 4p^r, p^{2r}+4)$, where $p$ is a prime greater than 3, $p\not \equiv 1\pmod 8,$ and the positive integers $a, n$ satisfy either $P(a)\,|\,n$ or $P(n)\nmid a$.
Key wordsJesmanowicz' conjecture    Pythagorean triple
 PACS: O156.1

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