关于Diophantine方程ax(x+1)…(x+z)=y(y+1)…(y+z)

On the Diophantine Equation ax(x+1)…(x+z)=y(y+1)…(y+z)

设a是大于1的正整数。本文运用Pell方程的基本性质证明:当a是平方数时,方程ax(x+1)…(x+z)=y(y+1)…(y+z)仅有有限多组正整数解(x,y,z)适合y-x=2;当a是非平方数时,该方程有无穷多组正整数解(x,y,z)适合y-x=2。

Let a be a positive integer with a〉1. In this paper, using some basic properties of Pell＇s equation, we prove that the equation ax（x＋1）…（x＋z）=y（y＋1）…（y＋z） has infinitely many positive integer solutions （x, y, z） with y-x=2 if and only ifa is not a square. K