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)
This paper starts with a review of Pell’s equation and continued fractions. It includes the following problems:
According to the records, it was presented to the St. Petersburg Academy on December 3, 1772.
- (section 18) If the formula Ax2 - 2Bxy + Cy2, 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 B2 - AC is a positive number and not a square.
- Given a formula Ax2 - Bxy + Cy2 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.
Originally published in Novi Commentarii academiae scientiarum Petropolitanae 18, 1774, pp. 218-244
Opera Omnia: Series 1, Volume 3, pp. 310 - 334
- Reprinted in Commentat. arithm. 1, 1849, pp. 570-583 [E454a]
- 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. 165-180 (1987).
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