Demonstration of a new method in the theory of arithmetic
According to Jacobi, a treatise with this title was read to the Berlin Academy on June 8, 1758. According to St. Petersburg Academy records, it was presented to that Academy on October 15, 1759.
Euler presents a third proof of the Fermat theorem, the one that lets us call it the Euler-Fermat theorem; this seems to be the proof that Euler likes best. He also proves that the smallest power xn that, when divided by a number N, prime to x, and which leaves a remainder of 1, is equal to the number of parts of N that are prime to n. In other words, the order of x modulo N is equal to the number of distinct aliquot parts of N.
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 8, pp. 74-104.
Opera Omnia Citation
Series 1, Volume 2, pp.531-555.