  We comment on two formal proofs of Fermat's sum of two squares theorem,
written using the Mathematical Components libraries of the Coq proof assistant.
The first one follows Zagier's celebrated one-sentence proof; the second
follows David Christopher's more recent proof relying on partition-theoretic
arguments. Both formal proofs rely on a general property of involutions of
finite sets, of independent interest. The proof technique consists for the most
part of automating recurrent tasks (such as case distinctions and computations
on natural numbers) via ad hoc tactics. With the same method, we also provide a
formal proof of another classical result on primes of the form $a^2 + 2 b^2$.
