The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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the successive solutions of the equation x2 - Dy2 = 1 are
deduced from the series of numbers which separate the
successive solutions of the given equation.
E. LEMOINE, "Resolution complete des equations indeterminees, x2 + 1 = 2y2; x2- = 2y2," Coimbra,
1893; also Jornal de sciencias mathematicas e astronomicas, vol. XI, p. 68, Coimbra, 1892.
K. SCHWERING, "Zerfallung der lemniskatischen Theilsungsgleichung in vier Factoren," Journal fur die reine
und angewandte Mathematik, vol. CX, p. 42, vol. CXII,
p. 37, Berlin, 1892, 1894.
The solution of the Pell equation is connected with the
theory of circle division and the theory of quadratic
remainders.
H. WEBER, "Ein Beitrag zur Transformationstheorie
der elliptischen Functionen mit einer Anwendung auf
Zahlentheorie," Mathematische Annalen, vol. XLIII,
p. 185, Leipzig, 1893.
The Pell equation is used to furnish the solutions of
certain modular equations called Schlafi'sche equations,
u = f(w) and v = f(nw), where n - 1 (mod 24) and is
prime and greater than 3. As an example of the connection of the Pell equation with elliptic functions the equation
x2 - 745y2 = 1
is discussed.
A. CAYLEY, C. E. BICKMORE, A. R. FORSYTH, A. LODGE,
J. J. SYLVESTER, "Tables connected with the Pellian
equation," Report of the British Association, vol. LXII,
p. 73, Nottingham, 1893, London, 1894, A. Cayley,
"Papers," vol. XIII, p. 430, Cambridge, 1897.
This table gives the fundamental solution of
x2 - Ay2 = - 1
or of
x2 - Ay2 = + 1
from
A = 1,001