6 is kind of cheating. It's a restatement of 45^2.
3^2 is the sum of the first three odd numbers. 4^2 is the sum of the first four odd numbers. 5^2 is the sum of the first five odd numbers.
Edit: sorry, don't mean to be a pill.
3^2 is the sum of the first three odd numbers. 4^2 is the sum of the first four odd numbers. 5^2 is the sum of the first five odd numbers.
Edit: sorry, don't mean to be a pill.
Obviously all the formula will be equivalent to each other. They are, by construction, all restatements of each other.
Actually, property 5) trivially implies 1) but also 2), as `(1+2+...+n)² = n²(n+1)²/4` and either n or n+1 must be divisible by 2 hence one of the squares divisible by 4 hence it is a product of squares. But also property 4) as `(1+2+...+n)² = 1³+2³+...+n³` (easy to show by induction).
You can prove it by induction; just expand (n(n+1)/2)² – (n(n-1)/2)², the result is n³.