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 Â± 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, ....
Original Source Citation
Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 11, pp. 78-93.
Opera Omnia Citation
Series 1, Volume 4, pp.255-268.