Proof of a theorem of Fermat that every prime number of the form 4n+1 is the sum of two squares
Euler probably already knew the "descent" step, as described in David Cox's Primes of the Form x2+ny2. Here, he does the "reciprocity" step by using the Euler-Fermat theorem to say that a4n−b4n is divisible by 4n+1. Then he factors it as (a2n+b2n)(a2n−b2n), and says that the second factor cannot be divisible by 4n+1; hence the first factor is divisible by 4n+1. As indexed by Eneström, this paper concludes on p. 13 of the Novi Commentarii, but Euler's work continues immediately into E242.
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 5, pp. 3-13.
Opera Omnia Citation
Series 1, Volume 2, pp.328-337.