The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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The Pell equation is here made serviceable in factorization.
A. CUNNINGHAM, "On the factorization of the Pellian
terms," British Association Report, p. 462, London, 1907.
J. SOMMER, "Vorlesung iiber Zahlentheorie," Leipzig,
1907.
In obtaining the units of a quadratic domain, f( m.),
where the fundamental number, m, is real, the solution
of the Pel] equation is necessary. Proof is given that
in the case of a real domain the fundamental unit can
always be found and from it the other units may be
obtained. This corresponds exactly to the proof of the
solvability of the equations
x2 - my = 1
and
x2 +x+ [ ( — ) ]2=1
(p. 102). This work has been revised and translated into
French by A. Levy, "Introduction a la theorie des
nombres algebriques," Paris, 1911.
B. NIEWENGLOWSKI, "Note sur les equations
x2 - ay2 = 1 et x2 -ay2 = -1,"
Bulletin de la Societe Mathematique de France, vol.
XXXV, p. 126, Paris, 1907.
The equation x2- ay2 = 1 represents in rectangular
axes an hyperbola. Integral points are defined as points
of this hyperbola having integral numbers for co6rdinates. If M and M1 are two integral points, the parallel
to the tangent at M1 drawn through M meets the hyperbola again in an integral point. Other similar properties
are proved for both equations.
P. A. MACMAHON, "The Diophantine equation
xn - Nyn = z," Proceedings of the London Mathematical Society, vol. V (2), p. 45, London, 1907.