Four squares formula

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.

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

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 proven 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 ${\displaystyle a_{k}}$ to ${\displaystyle -a_{k}}$, and/or any ${\displaystyle b_{k}}$ to ${\displaystyle -b_{k}}$.

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