不定方程6x(x+1)(x+2)(x+3)=13y(y+1)(y+2)(y+3)的正整数解研究

A research of the positive integer solution of the diophantine equation 6x(x+1)(x+2)(x+3)=13y(y+1)(y+2)(y+3)

不定方程是数论中的一个重要分支,利用递归序列,同余式方法等初等方法和Pell方程,证明了不定方程6x(x+1)(x+2)(x+3)=13y(y+1)(y+2)(y+3)仅有唯一正整数解(x,y)=(10,8),并给出了该方程的全部整数解.

In this article,by the mean of recurrence sequence,congruent form,Pell equation an Jacobian symbol,it is showed that(x,y)=(10,8)is the only positive integer solution of the diophantine equation 6x(x+1)(x+2)(x+3)=13y(y+1)(y+2)(y+3).Furthermore,the total integer solutions of the Diophantine equation x^(2)-78(y^(2)+3y+1)^(2)=-42 is obtained.