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:
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Originally published in Memoires de l'academie des sciences de St.-Petersbourg 5, 1815, pp. 3-23
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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
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