一个丢番图方程及其它的整数解

刘燕妮, 郭晓艳

数学学报 ›› 2010, Vol. 53 ›› Issue (5) : 853-856.

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数学学报 ›› 2010, Vol. 53 ›› Issue (5) : 853-856. DOI: 10.12386/A2010sxxb0095
论文

一个丢番图方程及其它的整数解

    刘燕妮1, 郭晓艳2
作者信息 +

A Diophantine Equation and Its Integer Solutions

    Yan Ni LIU1, Xiao Yan GUO2
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文章历史 +

摘要

研究丢番图方程 xy + yz + zx = 0 的可解性, 并求该方程的所有整数解.本文利用初等方法及整数的整除性质研究这一问题, 获得了彻底解决.即就是证明了方程 xy + yz + zx = 0有且仅有六组整数解 (x, y, z) = (-2, 1, 1), (1, -2, 1), (1, 1, -2), (1,-1,-2), (-1, -2, 1), (-2,1,-1).

 

Abstract

The main purpose of this paper is using the elementary method and the divisible properties of the integers to study this problem, and solve it completely. That is, we shall prove that the Diophantine equation xy + yz + zx = 0 has and only has six integer solutions. They are: (x, y, z) = (-2, 1, 1), (1, -2, 1), (1, 1, -2), (1,-1,-2), (-1, -2, 1), (-2,1,-1).

 

关键词

初等方法 / 不定方程 / 整数解

Key words

diophantine equation / elementary method / integer solutions

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刘燕妮, 郭晓艳. 一个丢番图方程及其它的整数解. 数学学报, 2010, 53(5): 853-856 https://doi.org/10.12386/A2010sxxb0095
Yan Ni LIU, Xiao Yan GUO. A Diophantine Equation and Its Integer Solutions. Acta Mathematica Sinica, Chinese Series, 2010, 53(5): 853-856 https://doi.org/10.12386/A2010sxxb0095

参考文献


[1] Kenichiro K., Comments and Topics on Smarandache Notions and Problems, Erhus University Press, USA, 1996.

[2] Smarandache F., Only Problems, Not Solutions, Chicago: Xiquan Publishing House, 1993.

[3] Tom M. A., Introduction to Analytic Number Theory, New York: Springer-Verlag, 1976.

[4] Zhang W. P., etc., Elementary Number Theory, Xi'an: Shaanxi Normal University Press, 2007 (in Chinese).

[5] Pan C. D., Pan C. B., Elementary Proof of the Prime Number Theorem, Shanghai: Shanghai Science and Technology Press, 1988 (in Chinese).

基金

国家自然科学基金资助项目(10671155);西北大学研究生自主创新基金(08YZZ30)

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