The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

95

Euler calculated the fundamental solutions of
x2 - Ay2 = 1
from A = 2 to A = 68 as published in the Commentaries1
of the St. Petersburg Academy, and to A = 99 as given
in his algebra.2 In the Novi Commentarii his table
occupies two pages and is entitled, "Tabula numerorum
p et q quibus fit pp = Iqq + 1 pro omnibus valoribus
numeri 1 usque ad 100." A line is drawn across all the
columns to denote the omission of the square numbers.
At the conclusion of the table solutions are given for the
numbers, 103, 109, 113, 157, 367.
In the first edition of Legendre's Theory of numbers,3
table XII at the end of the volume contains the solutions,
written in the form of a ratio m/n, of m2 - an2 = - 1 when
possible, otherwise of m2 - an2 = 1 for all non-square
values of a from 2 to 1,003. There is no mark to indicate
to which equation the solution, m, n, belongs. There are
errors4 in one or both the values, m, n, under the following
values of a: 133, 214, 236, 301, 307, 331, 343, 344, 355,
365, 397, 501, 526, 533, 613, 619, 629, 655, 664, 671, 694,
718, 732, 753, 771, 801, 806, 809, 851, 856, 865, 871, 878,
886, 944, 965, 995, 1,001. The most of these are corrected
in the third edition. In Legendre's second edition, table
X at the end of the volume contains the Pellian solutions
with the same arrangement as before but carried only to
the argument 135.
The Canon Pellianus of Degen5 was published in 1817.
1 L. Euler, Commentarii Academiae scientiarum imperialis Petropolitanae, 1732-3, vol. VI, p. 175, St. Petersburg, 1738 —"Commentaxii
arithmeticalvol. I, p. 4, St. Petersburg, 1849.
2 L. Euler, "Algebra," vol. II, p. 328, St. Petersburg, 1770; translated
with additions by J. L. Lagrange, vol. II, p. 133, Lyons, 1774.
3A. M. Legendre, "Essai sur la theorie des nombres," Paris, 1798.
2d ed., Paris, 1808, 3d ed., Paris, 1830.
4A. Cunningham, "On high Pellian factorisations," Messenger of
mathematics, vol. XXXV, p. 182, London, 1906.
5 C. F. Degen, "Canon Pellianus sive tabula simplicissimam aequationis
celebratissimae y2 = ax2 + 1 solutionem pro singulis numeri dati valoribus