## 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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