The Pell equation, by Edward Everett Whitford.

52

THE PELL EQUATION

in which the coefficient of the first term in parentheses
is 1, of the second the series of natural numbers, of the
third the triangular numbers, of the fourth the pyramidal
numbers (the sums of the first n triangular numbers), etc.
For computing these in the simplest manner Brouncker
gave the following method, in which yi, y2, y3,... are the
values of y:
Y1. = y1, Y2 = ZlyX, Y3 = z1i2 - Y1, Y4 = z 3 - Y2,
Y5 = Z1y4 - y3, ''
In order to make a complete reply to Fermat's challenge,
there was only needed a sure method for obtaining the
fundamental solution, and this Brouncker succeeded in
finding, as is set forth in the letters of Wallis under the
dates December 17, 1657, and January 30, 1658.1 Euler2
has also given the same solution in his algebra.
Stated in modern symbolism Brouncker's solution is
as follows.3 Suppose a solution of the equation
2 - Dy2 = 1
exists, and let this solution be T, U. Then
(1) T2 - DU2 = 1.
It is required to find T and U. If p is the integer next
smaller than /D, then it follows from the hypothesis that
pU < T < (p + 1)U,
and that
T = pU + vi,
1 "Commercium epistolicum," XVII, p. 797, and XIX, p. 804. "Oeuvres de Fermat," vol. III, p. 490, 503. The most difficult problem solved
here (p. 498), was
x2 - 433y2 = 1,
in which y has 19 figures.
2L. Euler, "Algebra," translated by J. Hewlett, 5th ed., p. 351, London, 1840. "Vollstandige Anleitung zur Algrabra," Part II, p. 226,
Lund, 1771.
3 H. J. S. Smith, "Report on the theory of numbers," British Association report, p. 292, London, 1861; "Collected mathematical papers,"
vol. I, p. 193, Oxford, 1894. H. Konen, "Geschichte der Gleichung
t2 - Du2 = 1," p. 39, Leipzig, 1901.