E26  Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus
(Observations on a theorem of Fermat and others on looking at prime numbers)
Summary:
Euler shows that the fifth Fermat number, 2^{25} +1 = 4 294 967 297, is not prime because it is
divisible by 641, though he does not give any clues about how he discovered this fact. He also tacks on a few "theorems"
but says that he does not yet know how to prove them.
According to the records, it was presented to the St. Petersburg Academy on September 26, 1732.
Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 6, 1738, pp. 103107

Opera Omnia: Series 1, Volume 2, pp. 1  5
 Reprinted in Comment. acad. sc. Petrop. 6, ed. nova, Bononiae 1743, pp. 98102 [E26a]
 Reprinted in Commentat. arithm. 1, 1849, pp. 13 [E26b]
 A handwritten French translation of this treatise can be found in the library of the observatory in
Uccle, near Brussels.
Documents Available:
 Original publication: E026
(in the Commentarii)
 David Zhao of the University of Texas has completed a parallel text translation of E26, which he has made available to the Euler Archive.
 Jordan Bell of the University of Toronto has also translated this article: E026
 Ian Bruce has translated this article, along with E54, into English.
 Euler's work on E26 was motivated by Fermat's claim that all integers of the form 2^{2n} + 1 are prime.
Fermat stated his claim in two letters to Frenicle in 1640.
Amanda Bergeron and David Zhao of the University of Texas have completed parallel text translations of these two letters, and have made them available to the Euler Archive:
 The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars.
Selected references we have found that discuss or cite E26 include:
 Calinger R., “Leonhard Euler: The first St Petersburg years (17271741).” Historia Mathematica, 23 (2), pp. 121166 (May 1996).
 Sandifer E., "Fermat's Little Theorem." How Euler Did It. (Published online
by the MAA.)
Return to the Euler Archive