Lagrange's Four-square Theorem - Proof Using The Hurwitz Integers

Proof Using The Hurwitz Integers

One of the ways to prove the theorem relies on Hurwitz quaternions, which are the analog of integers for quaternions. The Hurwitz quaternions consist of all quaternions with integer components and all quaternions with half-integer components. These two sets can be combined into a single formula

where E₀, E₁, E₂, E₃ are integers. Thus, the quaternion components a₀, a₁, a₂, a₃ are either all integers or all half-integers, depending on whether E₀ is even or odd, respectively. The set of Hurwitz quaternions forms a ring; in particular, the sum or product of any two Hurwitz quaternions is likewise a Hurwitz quaternion.

The generalized Euclidean algorithm identifies the greatest common right or left divisor of two Hurwitz quaternions, where the "size" of the remainder ρ is measured by the norm. The norm N(α) of a quaternion α is the nonnegative real number

,

where is the conjugate of α.

Since quaternion multiplication is associative, and real numbers commute with other quaternions, the norm of a product of quaternions equals the product of the norms:

.

The ring of Hurwitz quaternions is Euclidean, since for any quaternion with rational coefficients we can choose a Hurwitz quaternion so that by first choosing so that and then so that for 1, 2, 3. Then we obtain Therefore, the ring of Hurwitz quaternions is Euclidean and, as a consequence, also a unique factorization domain.

Since any natural number can be factored into powers of primes, the four-square theorem therefore holds for all natural numbers if it is true for all prime numbers. It is true for 2 = 02 + 02 + 12 + 12. To show this for an odd prime integer p, represent it as a quaternion (p, 0, 0, 0) and assume that it is not a Hurwitz prime; that is, it can be factored into two non-unit Hurwitz quaternions

p = αβ.

The norms of p, α, β are nonnegative integers such that

N(p) = p2 = N(αβ) = N(α)N(β).

But the norm has only two factors, both p. Therefore, if it can be factored into Hurwitz quaternions, then p is the sum of four squares

p = N(α) = a₀2 + a₁2 + a₂2 + a₃2.

Lagrange proved that any odd prime p divides at least one number of the form 1 + l2 + m2, where l and m are integers. The latter number can be factored in Hurwitz quaternions:

1 + l2 + m2 = (1 + l i + m j) (1 - l i- m j).

If p could not be factored in Hurwitz quaternions, it would be a Hurwitz prime number by definition. Then, by unique factorization, p would have to divide either 1 + l i + m j or 1 - l i- m j to form another Hurwitz quaternion. But for p>2, the number

1/p ± l/p i ± m/p j

is not a Hurwitz integer. Therefore, every p>2 can be factored in Hurwitz quaternions, and the four-square theorem holds.