Euler's four-square identity

An algebraic identity expressing the product of two sums of four squares as another sum of four squares. It establishes that numbers representable as sums of four squares remain so under multiplication.

Quaternion: A number system extending the complex numbers with three imaginary units and noncommutative multiplication. Quaternion norms turn the identity into the multiplicativity of squared length.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped analysis, number theory, and mechanics. The identity bears Euler's name and appears in his work on sums of squares.
Sum of four squares: An expression of a number as the sum of four squared quantities. The identity combines two such representations into another representation of the same kind.
Sum of squares: An expression formed by adding squared quantities. The identity is a multiplication rule for a particular four-term sum-of-squares expression.
Quaternion norm: For a quaternion, the sum of the squares of its four real coordinates. The identity is precisely the rule that quaternion norms multiply.
Diophantus: An ancient Greek mathematician known for problems involving integer solutions to algebraic equations. Questions about representing integers by squares reach back to Diophantine arithmetic.
Quaternion multiplication: The noncommutative multiplication operation on quaternions. Expanding quaternion products produces the four component expressions in the identity.
Polynomial identity: An equation between polynomials that holds for every allowed value of its variables. Euler's formula holds universally for its eight input variables.
Lagrange's identity: An identity equating the product of two sums of squares with a sum of squares built from pairwise combinations. Its two-square form is the lower-dimensional analogue of Euler's formula.
Fermat's theorem on sums of two squares: A theorem characterizing primes expressible as sums of two integer squares. It concerns the arithmetic family whose multiplication law is the two-square analogue.

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Degen's eight-square identityCompared with: It is the lower-dimensional quaternion analogue of Degen's formula.