An Application of Program Derivation Techniques to 18th-Century Mathematics

This note presents a derivation of one of the proofs given by Euler [4] of the famous theorem that every prime congruent to 1 modulo 4 is the sum of two squares—an observation commonly credited to Fermat but attributed to Girard [5] in [2]. The proof seems to be of methodological interest because it demonstrates that the techniques for program derivation developed by computing science have matured to a level where a proof that originally took many years to find may now be constructed with a minimum of invention. Incidentally, the proof also furnishes a counterexample to Dijkstra’s remark [3] that all known methods of factorization are nonconstructive. (It does so because writing a prime as x2+y2 is equivalent to factorizing it as (x + i · y) · (x− i · y) in Z[i].) Of course, this note does not provide any information on the way mathematics was actually done in the 18th century. In what follows, we shall depart from established mathematical notation in one respect. The assertion that x and y differ by a multiple of k will be written as