两类高次不定方程正整数解的讨论  

A DISCUSSION ON POSITIVE INTEGER SOLUTIONS TO TWO TYPES OF HIGH-ORDER INDEFINITE EQUATIONS

作　　者：高丽 GAO Li(School of Information Engineering,Xi’an Fanyi University,Xi’an,Shaanxi 710105,China)

机构地区：[1]西安翻译学院信息工程学院,陕西西安710105

出　　处：《井冈山大学学报（自然科学版）》2024年第6期23-26,42,共5页Journal of Jinggangshan University (Natural Science)

基　　金：国家自然科学基金项目(62261055);陕西省教育科学“十二五”规划项目(SGH22Y1334)。

摘　　要：本研究讨论了两类典型的高次不定方程求解问题,首先利用整除的性质、不定方程组求解等方法,对不定方程Mx(x+1)(x+2)(x+3)=Ny(y+1)(y+2)(y+3),在M=5^(2k),N=1时的正整数解进行研究,证明了这个不定方程不存在正整数解;其次利用递归序列、同余、Pell方程解的性质、勒让德符号等方法证明了不定方程x^(3)±1=Dy^(2)在D=1547时的情形,不定方程x^(3)-1=1547y^(2)仅有平凡整数解(1,0);不定方程x^(3)+1=1547y^(2)仅有平凡整数解(-1,0)。Indefinite equation is one of the important contents of number theory,In this paper,the solution of two kinds of typical high order indefinite equations was discussed.First,by using the property of division,the solution of the indefinite equation,etc.,the positive integer solution of Mx(x+1)(x+2)(x+3)=Ny(y+1)(y+2)(y+3),was studied at the condition of M=5^(2k),N=1,it is proved that there is no positive integer solution to this indefinite equation.Secondly,by using elementary methods such as recursive sequences,congruences,properties of solutions to the Pell equation,and Legendre symbols,it has been proven that an indefinite equation x^(3)-1=1547y^(2)only has trivial integer solutions(1,0)the indefinite equation x^(3)+1=1547y^(2)only has trivial integer solutions(-1,0)when D=1547 for the indefinite equation x^(3)±1=Dy^(2).

关 键 词：不定方程 正整数解 整除 同余 PELL方程 勒让德符号 

分 类 号：O156.4[理学—数学]