Prove that n(n+1) is never a square.

1. The problem statement, all variables and given/known data

Prove that n(n+1) is never a square for n>0

3. The attempt at a solution

n and n+1 are relatively prime because if they shared common factors then it should divide their difference but (n+1)-n=1 so 1 is their only common factor.

So the only possible way for n(n+1) to be a square is if n and n+1 are squares.

I will show that it is impossible to have perfect squares that are 1 apart.

Let x^2 and y^2 be squares that are 1 apart

so we have [itex] x^2-y^2=1=(x+y)(x-y) [/itex]

if x>y then x-y is at least 1 and x+y is bigger than 1 so this cant happen.