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, .

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