Euler's Four-Square Identity
Identity
The Four-Square Identity, credited to Leonhard Euler, states that for any eight complex numbers , we must have
In other words, the product of the sums of four squares is itself the sum of four squares.
Proof
First, let us expand the left-hand side of the identity:
Thus, we have
Now, let us expand the first square of the right-hand side of the identity, :
Therefore, we now also have
Next, let us expand the second square of the right-hand side of the identity, :