Abstract Fermat’s two squares theorem asserts that a prime one more than a multiple of 4 is a sum of two squares. There are many proofs of this gem in number theory, including a remarkable one-sentence proof by Don Zagier based on two involutions on a finite set built from such a prime. Applying the two involutions alternatively leads to an iterative algorithm to find the two squares for the prime. Moreover, a detailed analysis of the computation reveals that it is possible to jump through the iteration nodes, leading to a better hopping algorithm. Here is a formalisation of Zagier’s proof, deriving the involutions using windmill patterns. Theories developed for the formal proof are used to establish the correctness of both algorithms.

The two squares theorem of Fermat is a gem in number theory, with a spectacular one-sentence "proof from the Book". Here is a formalisation of this proof, with an interpretation using windmill patterns. The theory behind involves involutions on a finite set, especially the parity of the number of fixed points in the involutions. Starting as an existence proof that is non-constructive, there is an ingenious way to turn it into a constructive one. This gives an algorithm to compute the two squares by iterating the two involutions alternatively from a known fixed point.