E610 -- Novae demonstrationes circa divisores numerorum formae \(xx+nyy\)
(New demonstrations about the divisors of numbers of the form \(xx+nyy\))
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 \(4ni-2n+1\).
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]
- Original Publication: E610
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