E702  De novo genere quaestionum arithmeticarum pro quibus solvendis certa methodus adhuc desideratur
(English Translation of Title)
Summary:
Euler searches for all integers N such that the formulas A^{2} + B^{2} and
A^{2} + NB^{2} can both be squares at the same time. By putting A = xx  yy and
B = 2xy, the first expression becomes a square; to make the other one a square also, one takes the A^{2}
to be zz and obtains (x + y)/z ± xx/(zz), and the question reduces to finding values
for z such that N becomes an
integer. He finds, among the first 100 natural numbers, the following values for N that satisfy the problem: 7, 10, 11, 17, ….
Publication:

Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 11, 1798, pp. 7893

Opera Omnia: Series 1, Volume 4, pp. 255  268
 Reprinted in Commentat. arithm. 2, 1849, pp. 190197 [E702a]
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