## E744 -- De divisoribus numerorum in forma $$mxx + nyy$$ contentorum

(On divisors of numbers of the form $$mxx + nyy$$)

Summary:

First, Euler notes that divisors of $$xx + yy$$ ($$x$$ and $$y$$ relatively prime) must be of the form $$4N + 1$$, then that divisors of $$2xx + yy$$ must be of the form $$8N + 1$$ or $$8N + 3$$, and those of $$3xx + yy$$ are of the form $$12N + 1$$ or $$12N + 7$$. He generalizes this result by analyzing the divisors of $$mxx + nyy$$. Interestingly, one can state the congruence classes modulo $$4mn$$ which hold divisors of $$mxx + nyy$$ once one knows the value of the product $$mn$$, regardless of the particular values of $$m$$ and $$n$$. Euler presents a table of congruence classes modulo $$4mn$$ holding divisors of $$mxx + nyy$$ for small values of $$mn$$.

According to the records, was presented to the St. Petersburg Academy on June 1, 1778.

Publication:
• Originally published in Memoires de l'academie des sciences de St.-Petersbourg 5, 1815, pp. 3-23
• Opera Omnia: Series 1, Volume 4, pp. 418 - 431
• Reprinted in Commentat. arithm. 2, 1849, pp. 272-280 [E744a]
Documents Available:
• Original publication: E744
• English translation (Paul Bialek and Dominic Klyve): E744