关于丟番图方程x(x+1)(x+2)(x+3)=27y(y+1)(y+2)(y+3)

On the Diophantine Equation x(x+1)(x+2)(x+3)=27y(y+1)(y+2)(y+3)

先后运用了pell方程、勒让德符号,同余关系,递归序列、二次平方剩余,分类讨论的有关方法,并通过使用数学软件Mathematica进行计算,证明了以下结论:不定方程x(x+1)(x+2)(x+3)=27y(y+1)(y+2)(y+3)没有正整数解,并找出了该方程的全部16组整数解.

In this paper,Pell equation,Legendre symbol,congruence relation,recursive sequence,quadratic squareresidue,and related methods of classified discussion have been usedsuccessively,and by using mathematical software Mathematica to calculate,thefollowing conclusions have been proved:the indefinite equation x(x+1)(x+2)(x+3)=27y(y+1)(y+2)(y+3) has no positive integer solution,and all 16 groups ofinteger solutions of the equation have been found.