E715 -- De variis modis numeros praegrandes examinandi, utrum sint primi necne?
(On various ways of examining very large numbers, for whether or not they are primes)
Euler reviews the fact that if a number has two representations as axx +
byy, then it cannot be prime, demonstrates the theorem by applying it to
12091 = axx + byy for (x, y)=(40, 9) and (4, 33), and using this to factor 12091.
He proves that if M and N are of the same form, then so is MN. He presents the problem:
Given a formula axx + byy and a number that can be
written in that form in only one way, when can you conclude that the number must be prime?
This reduces to a problem in congruent (idoneal) numbers, and Euler gets to repeat his table of 65 such numbers.
Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 13, 1802, pp. 14-44
Opera Omnia: Series 1, Volume 4, pp. 303 - 328
- Reprinted in Commentat. arithm. 2, 1849, pp. 198-214 [E715a]
- A handwritten French translation of this treatise can be found in the library of the observatory in
Uccle, near Brussels.
- Original Publication: E715
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