求解不定方程x(x+1)(x+2)(x+3)=57y(y+1)(y+2)(y+3)

On the Diophantine Equation x(x+1)(x+2)(x+3)=57y(y+1)(y+2)(y+3)

不定方程是数论的一个重要分支,其中除了部分确定系数的多项式的解被解决了外,还存有很多问题有待被研究。鉴于此,对形如mx(x+1)(x+2)(x+3)=ny(y+1)(y+2)(y+3),(n,m)=1的不定方程,首次使用GP辅助软件进一步计算其不同形式的整数解。研究主要运用了Pell方程、递归数列、同余式及(非)平方剩余等一系列的证明方法,将不定方程转化成Pell方程。通过证明Pell方程的四个结合类,运用勒让德符号、同余式和递归数列,并借助Matlab软件和GP软件工具,完全搜索了四个结合类的解,证明了不定方程x(x+1)(x+2)(x+3)=57y(y+1)(y+2)(y+3)无正整数解。

Diophantine equation is an important branch of number theory.Besides the solution of polynomial with certain coefficients,there are still many problems to be studied.In view of this,for the first time,the GP software is used to further calculate its different forms of integer solutionsfor the Diophantine equation of the form mx(x+1)(x+2)(x+3)=ny(y+1)(y+2)(y+3).A series of proof methods,such as Pell equation,recursive sequence,congruence and(non)square residue,are used to transform the indefinite equation into Pell equation.By proving the four associative classes of Pell equation,using Legendre symbol,congruence and recursive sequence,and with the help of the Matlab software and the GP software tools,the solutions of the four associative classes are completely searched.It is proved that the Diophantine Equation x(x+1)(x+2)(x+3)=57y(y+1)(y+2)(y+3)has no positive integer solution.