欧拉四平方和恒等式

• ( (a₁b₁-a₂b₂-a₃b₃-a₄b₄)²+
• ( (a₁b₂+a₂b₁+a₃b₄-a₄b₃)²+
• ( (a₁b₃-a₂b₄+a₃b₁+a₄b₂)²+
• ( (a₁b₄+a₂b₃-a₃b₂+a₄b₁)²

欧拉在1748年5月4日寄给哥德巴赫的一封信中提到了这个恒等式。它可以用基本的代数来证明，在任何交换环中都成立。如果as和bs是实数，有一个更加简洁的证明：这个等式表达了两个四元数的积的绝对值就是它们绝对值的积的事实，就像婆罗摩笈多-斐波那契恒等式与复数的关系一样。

拉格朗日用这个恒等式来证明四平方和定理。

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

Algebraic identity

For any pair of quadruples from a commutative ring, the following expressions are equal: （公式略）

Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be verified with elementary algebra.

The identity was used by Lagrange to prove his four square theorem. More specifically, it implies that it is sufficient to prove the theorem for prime numbers, after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any aₖ -aₖ, and/or any bₖ -bₖ.

If the aₖ and bₖ are real numbers, the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers. This property is the definitive feature of composition algebras.

Hurwitz's theorem states that an identity of form, (a₁²+a₂²+a₃²+… +aₙ²)(b₁²+b₂²+b₃²+… +bₙ²)=c₁²+c₂²+c₃²+… +cₙ²

where the cᵢ are bilinear functions of the aᵢ and bᵢ is possible only for n = 1, 2, 4, or 8.

Proof of the identity using quaternions

Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b). This defines the quaternion multiplication rule a×b, which simply reflects Euler's identity, and some mathematics of quaternions. Quaternions are, so to say, the "square root" of the four-square identity. But let the proof go on:

Let α =a₁+a₂i+a₃j+a₄k and β =b₁+b₂i+b₃j+b₄k be a pair of quaternions. Their quaternion conjugates are α ^(*)=a₁-a₂i-a₃j-a₄k and β ^(*)=b₁-b₂i-b₃j-b₄k. Then A:=α α ^(*)=a₁²+a₂²+a₃²+a₄²

and B:=β β ^(*)=b₁²+b₂²+b₃²+b₄².

The product of these two is AB=α α ^(*)β β ^(*), where β β ^(*) is a real number, so it can commute with the quaternion α ^(*), yielding AB=α β β ^(*)α ^(*).

No parentheses are necessary above, because quaternions associate. The conjugate of a product is equal to the commuted product of the conjugates of the product's factors, so AB=α β (α β)^(*)=γ γ ^(*)

where γ is the Hamilton product of α and β: （公式略）

Then （公式略）

γ =r+u where r is the scalar part and u=⟨ u₁,u₂,u₃⟩ is the vector part, then γ ^(*)=r-u （公式略）

So, （公式略）

Pfister's identity

Pfister found another square identity for any even power:

If the cᵢ are just rational functions of one set of variables, so that each cᵢ has a denominator, then it is possible for all n=2^(m).

• Thus, another four-square identity is as follows: （公式略）
• where u₁ and u₂ are given by （公式略）
• Incidentally, the following identity is also true: u₁²+u₂²=(b₁²+b₂²)²(b₃²+b₄²)