## E559 -- Nova subsidia pro resolutione formulae $$axx + 1 = yy$$

(New assistance for solving the formula $$axx + 1 = yy$$)

Summary:

This describes Euler's solution of the Pell equation without the use of continued fractions. It opens with the biggy: if $$a = 61$$, then $$x = 226,\!153,\!980$$ and $$y = 1,\!766,\!319,\!049$$. Euler includes the following problems:
• Knowing a solution to $$app - 1 = qq$$, find numbers $$x$$ and $$y$$ that make $$axx + 1 = yy$$; i.e., find a relationship between the positive equation and the negative equation.
• Knowing $$app - 2 = qq$$, solve $$axx + 1 = yy$$.
According to the records, it was presented to the St. Petersburg Academy on September 23, 1773.

Publication:
• Originally published in Opuscula Analytica 1, 1783, pp. 329-344
• Opera Omnia: Series 1, Volume 4, pp. 76 - 90
• Reprinted in Commentat. arithm. 2, 1849, pp. 35-43 [E559a]
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