E610 -- Novae demonstrationes circa divisores numerorum formae xx + nyy

(New demonstrations about the divisors of numbers of the form xx + nyy )

Summary:

First, Euler presents some familiar theorems about quadratic residues. Then he presents two main theorems:

If n is of the form 4k + 1 or 4k + 2, then the only prime divisors of xx + nyy are of the form 4ni + 2n - 1.
If n is of the form 4k or 4k - 1, then the only prime divisors of xx + nyy are of the form 4ni - 2n + 1.

Publication:

Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 1, 1787, pp. 47-74
Opera Omnia: Series 1, Volume 4, pp. 197 - 220
Reprinted in Commentat. arithm. 2, 1849, pp. 159-173 [E610b]

Documents Available:

Original Publication: E610

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