E556 -- De criteriis aequationis \(fxx + gyy = hz^2\), utrum ea resolutionem admittat nec ne?

(On the criteria of whether equation \(fxx + gyy = hz^2\) 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, what values of those letters \(f\), \(g\) and \(h\) give equations that have solutions? He shows, given an \(f\), a \(g\) and three values of \(h\) for which there are solutions, how to construct a fourth value of \(h\), and calls it (section 12) a "most elegant theorem."

According to the records, it was presented to the St. Petersburg Academy on December 7, 1772.

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