English Title

On divisors of numbers of the form mxx + nyy

Authors

Leonhard Euler

Enestrom Number

744

Fuss Index

14

Original Language

Latin

Published as

Journal article

Published Date

1815

Written Date

1778

Content 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.

Original Source Citation

Mémoires de l'académie des sciences de St.-Petersbourg, Volume 5, pp. 3-23.

Opera Omnia Citation

Series 1, Volume 4, pp.418-431.

Record Created

2018-09-25

Included in

Mathematics Commons

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