E683  De singulari genere quaestionum Diophantearum et methodo maxime recondita eas resolvendi
(On a singular type of Diophantine questions and a most recondite method by which they are to be resolved)
Summary:
First, Euler notices that if N is of the form aa + nbb, then Nl is also of that form. He demonstrates how to determine the forms with a few examples. Then he poses the general problem, N = aa + nbb, to find the powers of N that make Nl = xx + n, and, in general, how to minimize the value of y in a value xx + nyy.
Publication:

Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 9, 1795, pp. 318

Opera Omnia: Series 1, Volume 4, pp. 221  234
 Reprinted in Commentat. arithm. 2, 1849, pp.174182 [E683a]
Documents Available:
 Original publication: E683
Return to the Euler Archive