E54 -- Theorematum quorundam ad numeros primos spectantium demonstratio
(A proof of certain theorems regarding prime numbers)
(based on the abstract of David Zhao's English translation)
This paper presents the first proof of the Euler-Fermat theorem, also known as Fermat's Little Theorem, that ap-1 ≡ 1 mod p
for all a relatively prime to p. Euler begins by showing that 2p-1 ≡ 1 mod p for p ≠ 2, after which he
shows that 3p-1 ≡ 1 mod p for p ≠ 3. He then concludes that the formua holds for all a relatively prime to p.
the records, it was presented to the St. Petersburg Academy on August 2, 1736.
Originally published in Commentarii academiae scientiarum Petropolitanae 8, 1741, pp. 141-146
Opera Omnia: Series 1, Volume 2, pp. 33 - 37
- Reprinted in Comment. acad. sc. Petrop. 8, ed. nova, Bononiae 1752, pp. 127-132 [54a]
- Reprinted in Commentat. arithm. 1, 1849, pp. 21-23 [54b]
- A handwritten French translation of this treatise can be found in the library of the observatory in
Uccle, near Brussels.
- 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.
- In addition to this, Zhao and Amanda Bergeron of the University of Texas have completed parallel text translations of two letters in the Fermat-Frenicle correspondence, and have made them available to the Euler Archive:
- Ian Bruce has translated this article, along with E26, into English.
Return to the Euler Archive