E454  De resolutione irrationalium per fractiones continuas, ubi simul nova quaedam et singularis species minimi exponitur
(On the resolution of irrationals by continued fractions, where a certain minor new and singular type is set forth)
Summary:
This paper starts with a review of Pell’s equation and continued fractions. It includes the following problems:
 (section 18) If the formula Ax^{2}  2Bxy + Cy^{2}, in
case x = a and y = b, produces the value c, to find an
infinity of other values for x and y that produce that same value c, assuming
that the quantity B^{2}  AC is a positive number and not a square.
 Given a formula Ax^{2}  Bxy + Cy^{2} in which
BB  AC is a positive number and not a square, to find values for the letters
x and y that produce the minimum value in this formula.
According to the records, it was presented to the St. Petersburg Academy on December 3, 1772.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 18, 1774, pp. 218244

Opera Omnia: Series 1, Volume 3, pp. 310  334
 Reprinted in Commentat. arithm. 1, 1849, pp. 570583 [E454a]
Documents Available:
 Original Document: E454
 German Translation (Alexander Aycock and Arseny Skryagin): E454
 The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E454 include:
 Bremner A., “On Euler's quartic surface.” Mathematica Scandinavica, 61 (2), pp. 165180 (1987).
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