## 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)

Summary:

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.

Publication:
• Originally published in Opuscula Analytica 1, 1862, pp. 211-241
• Opera Omnia: Series 1, Volume 4, pp. 1 - 24
• Reprinted in Commentat. arithm. 2, 1849, pp. 556-569 [E556a]
Documents Available:
• Original Publication: E556