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 = 226153980 and y = 1766319049. 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:
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Originally published in Opuscula Analytica 1, 1783, pp. 329-344
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Opera Omnia: Series 1, Volume 4, pp. 76 - 90
- Reprinted in Commentat. arithm. 2, 1849, pp. 35-43 [E559a]
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