The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

117

F. LANDRY, "Cinquieme memoire sur la theorie des
nombres," Paris, 1856.
This article contains a theorem in relation to the solution of x2 - Ay2 = rm when A + r = a2. There are
applications to the methods of Lagrange and Gauss.
M. A. STERN, "Zur Theorie der periodischen Kettenbriiche," Journal fiur die reine und angewandte Mathematik, vol. LIII, p. 1, Berlin, 1857.
This article contains a hundred pages on the solution of
the Pell equation by the aid of continued fractions, closing
with what the author calls a Pellian table in which the
numbers A(< 1,000) are classified.
A. CAYLEY, "Note sur l'6quation x2 - Dy2 = - 4,
D - 5 (mod 8)," Journal fur die reine und angewandte
Mathematik, vol. LIII, p. 369, Berlin, 1857.
The fundamental solution of the equation x2 - Dy2 = 4
is deduced from the fundamental solution of the equation
t2- Du2 = -4 by means of x + y-JD =~ (t + u /D)2,
giving x = t2 + 2, y = tu. The fundamental solution of
the equation x2 - Dy2 = 1 is obtained from the fundamental solution of the equation T2 - DU2 = 4 by means of
x +y D= I(T + UD)3,
giving x =(T3 - 3T), y = (T2 -1) U. A table of
solutions of the equation x2 - Dy2 = =- 4, D _ 5 (mod 8)
is given for D = 5 to D = 997. Where such solutions
are possible they are much smaller than for the equation
x2 - Dy2 = 1.
G. C. GERONO, "Resoudre en nombres entiers l'6quation
x2- ny2 = 1, dans laquelle on suppose que n represente
un nombre entier, positif, non carre," Nouvelles annales
de mathematiques, vol. XVII, p. 122, 153, Paris, 1859.
J. F. KONIG, "Zerlegung der Gleichung x2 - fgy2 = = 1
in Faktoren," Archiv der Mathematik und Physik, vol.
XXXIII, p. 1, Griefswald, 1859.