The Pell equation, by Edward Everett Whitford.

114

THE PELL EQUATION

of an equation of finite differences of the form
y" -2my' + y = 0.
P. BARLOW, "An elementary investigation of the
theory of numbers," p. 294, London, 1811.
There are fifteen theorems on the Pell equation; and
the fundamental solutions of x2 - Ny2 = 1 from N = 2
to N = 102 are given.
P. BARLOW, "New mathematical tables," p. 266,
London, 1814.
General formulas are given for the solution of the
equations
x2- ay2 = z2, x2 Ny2 = 1, x y2Ny = A.
G. PALETTI, "Risilutione dell' equazione generale
completa di secondo grado a tre indeterminate," Rome,
1820.
The general solution of the indeterminate equation of
the second degree in three unknowns is made to depend
on the Pell equation.
E. F. A. MINDING, "Observatio pertinens ad solutionem
aequationum indeterminatarum secundi gradus," Journal
fur die reine und angewandte Mathematik, vol. VII, p.
140, Berlin, 1831.
P. N. C. EGEN, "Handbuch der allgemeinen Arithmetik,"
Part I, p. 457, Berlin, 1833, Part II, p. 467, Berlin, 1834.
Egen gives the 121 values of A < 1,000 for which
x2 - Ay2 = - 1
is solvable.
STERN, "Theorie der Kettenbruiche und ihre Anwendung," Journal fiir die reine und angewandte Mathematik,
vol. XI, p. 277, Berlin, 1834.
This is one of seven lengthy articles on this subject.
The applications to the Pell equation are contained on
pages 327-341. There is a table of continued fractions
for forty-two general forms.