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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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摘要 设 $ (a,b,c) $为本原的商高数组, 满足$ a^2+b^2=c^2$且 $2\,|\,b$. 1956年, Jesmanowicz猜想: 对任给的正整数$n$, 丢番图方程$ (na)^x+(nb)^y=(nc)^z$ 仅有正整数解 $x=y=z=2$. 令 $P(n)$表示$n$的所有不同素因子乘积.对商高数组 $ (a, b, c)=( p^{2r}-4, 4p^r, p^{2r}+4)$, 其中 $p$ 为大于3的素数且 $p\not \equiv 1\pmod 8$, 本文证明在条件 $P(a)\,|\,n $ 或者 $ P(n)\nmid a$下, Jesmanowicz 猜想成立.
关键词 Jesmanowicz 猜想商高数组    
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  
通讯作者: E-mail: cfjsz@126.com   
[1] Deng, M.J., A note on the Diophantine equation $(na)^x+(nb)^y=(nc)^z$, Bull. Aust. Math. Soc., 2014, 89(2): 316-321.
[2] Fu, C.Y. and Deng, M.J., On the Dionphantine $(n(7^{2r}-4))^x+( 4 n\cdot 7^r)^y=(n(7^{2r}+4))^z$, J. Nat. Sci. Heilongjiang Univ., 2015, 32(5): 596-599 (in Chinese).
[3] Jesmanowicz, L., Several remarks on Pythagorean numbers, Wiadom. Mat. (2), 1955/1956, 1: 196-202.
[4] Lu, W.T., On the Pythagorean numbers $4n^2-1, 4n, 4n^2+1$, Sichuan Daxue Xuebao}, 1959, 5(2): 39-42.
[5] Ma, M.M. and Wu, J.D., On the Diophantine equation $(an)^x+(bn)^y=(cn)^z$, Bull. Korean Math. Soc., 2015, 52(4): 1133-1138.
[6] Miyazaki, T., Generalizations of classical results on Jesmanowicz' conjecture concerning Pythagorean triples, J. Number Theory, 2013, 133(2): 583-595.
[7] Miyazaki, T., A remark on Jesmanowicz' conjecture for the non-coprimality case, Acta Math. Sin. (Engl. Ser.), 2015, 31(8): 1255-1260.
[8] Miyazaki, T., Yuan, P.Z. and Wu, D.Y., Generalizations of classical results on Jesmanowicz' conjecture concerning
Pythagorean triples II, J. Number Theory, 2014, 141: 184-201.
[9] Sierpinski, W., On the Diophantine equation $3^x+4^y=5^z$, Wiadom. Mat. (2), 1955/56, 1: 194-195.
[10] Tang, M. and Weng, J.X., Jesmanowicz' conjecture with Fermat numbers, Taiwanese J. Math., 2014, 18(3): 925-930.
[11] Terai, N., On Jesmanowicz' conjecture concerning primitive Pythagorean triples, J. Number Theory, 2014, 141: 316-323.
[12] Yang, H. and Fu, R.Q., A note on Jesmanowicz' conjecture concerning primitive Pythagorean triples, J. Number Theory, 2015, 156: 183-194.
[13] Zhang, X.W. and Zhang, W.P., The exponential Diophantine equation $((2^{2m}-1)n)^x +(2^{m+1}n)^y = ((2^{2m} +1)n)^z$,Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 2014, 57(3): 337-344.
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