The Pell equation, by Edward Everett Whitford.

96

THE PELL EQUATION

Table I contains the solutions of y2 - ax2 = 1, for all
non-square values of a from a = 1 to a = 1,000. Table
II contains solutions of y2 - ax2 = - 1, when possible,
within the same range of values of a; but though the
title of the table says that the solutions are given when
possible, the solutions for y2 - ax2 = - 1 when a is of
the form a 2 + 1 are omitted. In this case y = a, x = 1
would be a solution. It should also be noted that the
usual symbols y and x are transposed. Besides the solutions the tables of Degen also give the elements of the
continued fractions which lead to them, and this practice
is continued in the tables in the British Association
Reports. The following corrections1 should be made:
a = 238, for x = 1,756 read x = 756;? =437, for
y = 4,499 read y = 4,599; a = 672, for'y = 327 read
y = 337; a = 751, the last seven figures of! "y uld be.. 4,418,960; a = 823, insert 47 after 235,170 in the
value of y; a = 919, the last fourteen figures of x should
be 36,759,781,499,589; a = 945, for y = 27,551 read y =
275,561; a = 951, for y = 22,420,806 read y = 224,208,076.
In Legendre's third and fourth editions3 the table is
nearly the same as in the first edition, but the values of
x and y are printed in the form of a ratio x: y, and instead
of being printed in one list they are printed in several
according to the number of figures in the numbers x and y.
This table also has the inconvenience of the former table
in that there is no distinction between the solutions of
x2- Ny2 = 1 and x2 - Ny2 = - 1. The following corrigenda have been noted: N = 94, read x = 2,143,295;
N = 116, read x = 9,801; N = 149, read y = 9,305;
N = 308, read x = 351; N = 479, read y = 136,591; N
ab 1 usque ad 1000 in numeris rationalibus iisdemque integris exhibens,"
Copenhagen, 1817.
1 A. Cunningham, op. cit., p. 183.
2 C. F. Degen, op. cit., p. 112.
3 Legendre, "Theorie des nombres," 3d ed., Paris, 1830, 4th ed., Paris,
1900.