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