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 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.
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. 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.
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.
- 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.
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