Lagrange's Four Square Theorem

Theorem
Every positive integer can be expressed as a sum of four squares.

Proof
From Product of Sums of Four Squares it is sufficient to show that each prime can be expressed as a sum of four squares.

The prime number $$2$$ certainly can: $$2 = 1^2 + 1^2 + 0^2 + 0^2$$.

Now consider the odd primes.

Suppose that some multiple $$m p$$ of the odd prime $$p$$ can be expressed as:
 * $$m p = a^2 + b^2 + c^2 + d^2, 1 \le m < p$$.

If $$m = 1$$, we have the required expression.

If not, then after some algebra we can descend to a smaller multiple of $$p$$ which is also the sum of four squares:
 * $$m_1 p = a_1^2 + b_1^2 + c_1^2 + d_1^2, 1 \le m_1 < m$$.

Next we need to show that there really is a multiple of $$p$$ which is a sum of four squares.

From this multiple we can descend in a finite number of steps to $$p$$ being a sum of four squares.

Also see
Waring's Problem