The Pell equation, by Edward Everett Whitford.

152

THE PELL EQUATION

has the convergents Pa/Qa (a = 1, 2, 3,.*), the relation
(1 - 1)a = (- )la-(kQ- Pa)
exists only for
k = 2.
G. CANDIDO, "Sulle equazioni x2 - ay2 = zx, x2 - ay2
= b5," Giornale di matematiche di Battaglini, vol.
XLIII, vol. XII (2), p. 93, Naples, 1905.
This alludes to question 461 in L'Intermediare des
mathematiciens, vol. II, p. 308, Paris, 1895.
M. ELPHINSTONE, "History of India," p. 142, Note 16,
9th ed., E. B. Crowell, editor, London, 1905.
The editor compares the work of Diophantus with that
of Brahmagupta and the first solution of Lord Brouncker
with that of Bhaskara.
R. W. D. CHRISTIE, "Solve generally, in integers, the
Diophantine equation y2 = av2 + avy + 1, where a is
arbitrary," Mathematical questions from the Educational
Times, vol. X (2), p. 24, London, 1906.
There are solutions by the proposer, by A. Cunningham
and by A. Holm, but all involve the Pell equation.
A. HOLM, "If x = p, y = q, is any particular solution
of x2 - Cy2 = 4 D, where C is a positive integer not a
perfect square and D any integer, then all the [positive]
integral solutions are furnished by
x - y ~C = (p - q C)(r - s C)n,
where n is any integer positive or negative and x = r,
y = s, any particular solution of the equation
2 - Cy2 1,"
Mathematical questions from the Educational Times,
vol. X (2), p. 29, London, 1906.
This theorem will often enable one to dispense with
Lagrange's chain of reductions for the case when D > ~/C.
The solution is by B. Krishnamachari.