n(n+1)(n+2)(n+3) cannot be a square
Uniqueness of prime factors for a given number
The Attempt at a Solution
I'm not sure but I think I've proved a stronger case for how product of consecutive numbers cannot be squares. I dunno whether it is right
The product of two consecutive numbers are coprime so cannot be square
in the product of three consecutive numbers eg n(n+1)(n+2), two of those consecutive numbers are divisible by 2 and not a square number but the other isn't divisible by 2 . If that other is a perfect square, then also the product of a square and a nonsquare number cannot be square .Thus 3 consecutive numbers cannot form a perfect square
Now consider 4 numbers n(n+1)(n+2)(n+4). There are 3 consecutive numbers such that their products are divisible by 3 but are not a perfect square . The other number isn't divisible by 3 therefore product of 4 consecutive numbers cannot be perfect square . Thus so on and forever.