Yet another Proof of an old Hat

Every odd prime number p can be written in exactly (p+1)/2 ways as a sum ab + cd with min(a, b) > max(c, d) of two ordered products. This gives a new proof Fermat’s Theorem expressing primes of the form 1 + 4N as sums of two squares . Theorem 0.1. For every odd prime number p there exist (p+1)/2 ordered quadruplets (a, b, c, d) in N such that p = a·b+c·d and min(a, b) > max(c, d). As a consequence we obtain a new proof of the following result discovered by an old rascal who did not want to spoil his margins and left the proof to another chap who had no such qualms. Corollary 0.2. Every prime number of the form 1 + 4N is a sum of two squares. Proof of Corollary 0.2. If p is a prime-number congruent to 1 (mod 4), the number (p + 1)/2 of solutions (a, b, c, d) defined by Theorem 0.1 is odd. The involution (a, b, c, d) 7−→ (b, a, d, c) has thus a fixed point (a, a, c, c) expressing p as a sum of two squares. Corollary 0.2 has of course already quite a few proofs. Some are described in the entry “Fermat’s theorem on sums of two squares” of [2]. The author enjoyed also the account given in [1]. The set Sp of solutions defined by Theorem 0.1 is invariant under the action of Klein’s Vierergruppe with non-trivial elements acting by (a, b, c, d) 7−→ (b, a, c, d), (a, b, d, c), (b, a, d, c) . The following tables list all elements (a, b, c, d) with a, b, c, d decreasing together with the size ♯(O) of the corresponding orbit under Klein’s Vier