E598  De insigni promotione scientiae numerorum
(English Translation of Title)
Summary:
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:
 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.
Then a big theorem answers everything.
According to the records, it was presented to the St.
Petersburg Academy on October 26, 1775.
Publication:

Originally published in Opuscula Analytica 2, 1785, pp. 275314

Opera Omnia: Series 1, Volume 4, pp. 163  196
 Reprinted in Commentat. arithm. 2, 1849, pp. 140158 [E598a]
Documents Available:
 Original Publication: E598
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