Three proofs of Lagrange's Theorem in Hardy and Wright

Hardy and Wright's first proof of Lagrange's Theorem [HardyWright, §20.5] goes back to Euler. The key idea in the proof is to reduce representability of integers to representability of primes.

Lemma. The set of representable integers is closed under multiplication.

Sketch of Proof. Using elementary algebra, show that

$$(a^2 + b^2 + c^2 + d^2)(w^2 + x^2 + y^2 + z^2) = A^2 + B^2 + C^2 + D^2$$

where

$$A = aw + bx + cy + dz,$$

$$B = ax - bw + cz - dy,$$

$$C = ay - cw + dx - bz,$$

and

$$D = az - dw + by - cx.$$

After establishing this lemma, it suffices to prove that every positive prime is representable. The proof on page 302 starts with a lemma (Theorem 87, page 70) which says that some positive multiple of an odd prime $p$ is representable, or more specifically:

Lemma. If $p$ is an odd prime, then there are integers $x$, $y$ and $m$ with $0<m<p$ such that

$$1 + x^2 + y^2 = mp.$$

The basic idea in proving that every odd prime is representable is to show that with four squares, the smallest representable multiple of prime $p$ is $p$ itself. To be complete, one also needs to verify that $1$ and $2$ are also representable.

Hardy and Wright give a second proof [HardyWright, §20.9] of Lagrange's Theorem going back to a 1919 study by Hurwitz of integer quaternions. Since a representation of an integer as the sum of four squares can be viewed as the norm of an integer quaternion, Lagrange's Theorem is equivalent to the statement that every positive integer is the norm of some integer quaternion. For each integer $z$, Jacobi's Theorem counts the number of integer quaternions with norm $z$.

Hardy and Wright also give a more elementary elliptic functions (without elliptic functions) proof of Jacobi's Theorem using theta functions in sections §20.11 and §20.12. This proof goes back to Ramanujan.