how do i prove that the product of four adjacent positive integers cannot be a perfect square number?

hearts123 Nov 24, 2018

#1**+1 **

We have that

(a - 1) a ( a + 1) (a + 2) for a ≥ 2 where a is an integer

(a - 1) ( a + 1) a (a + 2)

(a^2 - 1) ( a^2 + 2a)

If this were a perfect square....both factors would have to be equal

But...the first factor will always be < the second factor for any a ≥ 2

So....this cannot be a perfect square

CPhill Nov 24, 2018