E598 -- De insigni promotione scientiae numerorum
(English Translation of Title)
Euler begins by noting Lagrange’s work on the divisors of numbers of the form Btt + Ctu + Duu and his
contribution to the "knowledge" or "science" of numbers. He includes the following problems:
Then a big theorem answers everything.
- Find the divisors of pp + nqq.
- Transform frr + grs + hss, in which 4fh - gg = 4n, into a diferent form f’tt + g’tu+h’uu,
in which g’ < f’ and g’ < h’, and still maintaining the property 4f’h’ - g’g’ = 4n.
- Find all the prime divisors of numbers in the form pp + nqq, where p and q are relatively prime with
respect to n.
- Find all the prime divisors of numbers in the form pp - nqq, where p and q are relatively prime
with respect to n.
According to the records, it was presented to the St.
Petersburg Academy on October 26, 1775.
Originally published in Opuscula Analytica 2, 1785, pp. 275-314
Opera Omnia: Series 1, Volume 4, pp. 163 - 196
- Reprinted in Commentat. arithm. 2, 1849, pp. 140-158 [E598a]
- Original Publication: E598
Return to the Euler Archive