E559 -- Nova subsidia pro resolutione formulae \(axx + 1 = yy\)
(New assistance for solving the formula \(axx + 1 = yy\))
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:
According to the records, it was presented to the St.
Petersburg Academy on September 23, 1773.
- 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 \(axx + 1 = yy\).
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]