丢番图方程y(y+1)(y+2)(y+3)=n^2 x(x+1)(x+2)(x+3)解的研究

On the Diophantine equation y(y+1)(y+2)(y+3)=n 2x(x+1)(x+2)(x+3)

设n>1是正整数,利用Pell方程的正整数解的一组恒等式和高次丢番图方程的结果,研究了丢番图方程y(y+1)(y+2)(y+3)=n^2x(x+1)(x+2)(x+3)的正整数解(x,y),分别在2|/n,3|x的情形下和n不同素因数的个数不超过2的情形下,证明了该方程没有正整数解(x,y).

Let n be a positive integer,n 1. The positive integer solution of the Diophantine equation y（ y ＋ 1）（ y＋ 2）（ y ＋ 3） = n^2x（ x ＋ 1）（ x ＋ 2）（ x ＋ 3） is investigated,with identical equalities of positive integer solutions of the Pell equation and the results of the high-order Diophantine equation. When 2 ｜/n,3 ｜ x,the fact that the Diophantine equation has no positive integer solution is proved in this paper. And the same result is also proved under the condition that the number of different prime factors of n is no more than 2.