E256  Specimen de usu observationum in mathesi pura
(Example of the use of observation in pure mathematics)
Summary:
Euler attempts to explain his thinking process by working out problems in quadratic forms in great detail. Euler notes that numbers of the form 2aa + bb are closed under multiplication. He develops more properites of such numbers and proves some of them. Then he uses the properties to show that 67579 = 2(87)^{2} + 229^{2} is prime, and that 40081 isn’t.
According to C. G. J. Jacobi, a treatise with this title was read to the Berlin Academy on
November 22, 1753.
According to the records, it was presented to the St. Petersburg Academy on
September 30, 1754.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 6, 1761, pp. 185230

Opera Omnia: Series 1, Volume 2, pp. 459  492
 Reprinted in Commentat. arithm. 1, 1849, pp. 174192 [E256a]
 A handwritten French translation of this treatise can be found in the library of the observatory in
Uccle, near Brussels.
Documents Available:
 Original publication: E256
 E256 is discussed in Ed Sandifer's How Euler Did It August 2005
and January 2006 columns published online by the MAA.
 Other works that cite this paper include:
Return to the Euler Archive