A detour through complex numbers

A classic example of this kind of adventure is the proof of the following result: if two integers are each a sum of four squares, so is their product.

Using complex numbers in this proof will make our task considerably easier! Suppose that the integers A and B are written A = a² + b² + c² + d² and B = p² + q² + r² + s². It will suffice to introduce the complex numbers z = a + ib, t = c + id, u = p + iq and v = r + is so that we can write

which is nothing other than ${\mid z\bar{u}+t\bar{v}\mid }^2+{\mid zv-tu\mid }^2\backslash left|z\backslash overline\{u\}+t\backslash overline\{v\}\backslash right|^\{2\}+\backslash left|zv-tu\backslash right|^\{2\}$ which is indeed a sum of four squares, each of the two terms being a sum of two squares. This very convenient result is used in particular in the proof of Lagrange's four-square theorem: it allows us to prove that every integer is a sum of four squares using only prime numbers.

Another, less well-known, example of an advantageous detour through complex numbers is the one cited by our colleagues from Quebec, Alain Desparois and Paul Guertin (Bulletin de l'Association mathématique du Québec, March 2017), when they look into n-similar functions, that is, functions that are equal to their n^th derivative, without being equal to any of their first n – 1 derivatives. The exponential function is obviously 1-similar, x $\mapsto \backslash mapsto$ exp (–x) is 2-similar, the sine function is 4-similar. How can we find real functions of a real variable that are 3-similar?