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 A2 + B2 and
A2 + NB2 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 A2
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. 78-93
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Opera Omnia: Series 1, Volume 4, pp. 255 - 268
- Reprinted in Commentat. arithm. 2, 1849, pp. 190-197 [E702a]
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