## E271 -- Theoremata arithmetica nova methodo demonstrata

(Demonstration of a new method in the theory of arithmetic)

Summary:

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 $$x^n$$ that, when divided by a number $$N$$, prime to $$x$$, and that leaves a remainder of 1, is equal to the number of parts of $$N$$ that are prime to $$n$$, that is to say, the number of distinct aliquot parts of $$N$$.

According to C. G. J. Jacobi, a treatise with this title was read to the Berlin Academy on June 8, 1758.

According to the records, it was presented to the St. Petersburg Academy on October 15, 1759.

Publication:
• Originally published in Novi Commentarii academiae scientiarum Petropolitanae 8, 1763, pp. 74-104
• Opera Omnia: Series 1, Volume 2, pp. 531 - 555
• Reprinted in Commentat. arithm. 1, 1849, pp. 274-286 [E271a]
• A handwritten French translation of this treatise can be found in the library of the observatory in Uccle, near Brussels.
Documents Available:
• Original publication: E271
• German translation (Artur Diener and Alexander Aycock): E271

• Other works that cite this paper include:
• Dickson
• Rudio
• Andre Weil, Number Theory