E54  Theorematum quorundam ad numeros primos spectantium demonstratio
(A proof of certain theorems regarding prime numbers)
Summary:
(based on the abstract of David Zhao's English translation)
This paper presents the first proof of the EulerFermat theorem, also known as Fermat's Little Theorem, that a^{p1} ≡ 1 mod p
for all a relatively prime to p. Euler begins by showing that 2^{p1} ≡ 1 mod p for p ≠ 2, after which he
shows that 3^{p1} ≡ 1 mod p for p ≠ 3. He then concludes that the formua holds for all a relatively prime to p.
According to
the records, it was presented to the St. Petersburg Academy on August 2, 1736.
Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 8, 1741, pp. 141146

Opera Omnia: Series 1, Volume 2, pp. 33  37
 Reprinted in Comment. acad. sc. Petrop. 8, ed. nova, Bononiae 1752, pp. 127132 [54a]
 Reprinted in Commentat. arithm. 1, 1849, pp. 2123 [54b]
 A handwritten French translation of this treatise can be found in the library of the observatory in
Uccle, near Brussels.
Documents Available:
 Original publication: E054 (in the Commentarii)
 David Zhao of the University of Texas has completed a parallel text translation of E54, which he has made available to the Euler Archive.
 Ian Bruce has translated this article, along with E26, into English.
Return to the Euler Archive