The Pell equation, by Edward Everett Whitford.

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THE PELL EQUATION

in which m is the fundamental solution for p, r, and s
the last found values for p and q, and r' and s' the next
preceding values. Dr. Hart claims that this method after
two solutions have been obtained is the simplest that
has been yet devised for the succeeding solutions.
W. SCHMIDT, "Uber die Auflisung der Gleichung
t2 - Du2 = - 4,
wo D eine positive ungerade Zahl und kein Quadrat ist,"
Zeitschrift ffir Mathematik und Physik, vol. XIX, p. 92,
Leipzig, 1874.
The solution is based upon the theory of quadratic
forms. The example given is
2 - 61y2 = 4.
M. COLLINS, A. M. NASH, "Prove that the equation
x2 + Dm = (N2 + D)y2 is always possible in rational
numbers for x and y when N and D are rational, and m is
an odd integer, and that x and y can be found in integers
when N and D are integers," Mathematical questions
from the Educational Times, vol. XXII, p. 23, London,
1875.
x= = NDn, y = D D, where m = 2n + 1.
S. TEBAY, "Solution of a question," Mathematical
questions from the Educational Times, vol. XXIII,
p. 30, London, 1875.
In order that nt2 = k be a square it is necessary that
t = a + - n- (++ r r) {an)(qr -- - -- ) m(fqr + 7-r)}
where
v = p + qn',
p, q, the smallest solution of
X2 - ny2 = 1,
and a any particular value of t.