On the criteria of whether equation fxx + gyy = hz2 admits a resolution or not
Euler's motivating example is that xx+yy = 2zz has solutions but that xx+yy = 3zz has no solutions. The question is, for what values of f, g and h will fxx+gyy = hzz have solutions? He shows that, given single values for f and g and three values of h for which there are solutions, one may construct a fourth value for h, and calls it (section 12) a "most elegant theorem."
Original Source Citation
Opuscula analytica, Volume 1, pp. 211-241.
Opera Omnia Citation
Series 1, Volume 4, pp.1-24.