The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

147

The conditions are deduced from the theory of the Pell
equations x2 - Ny2 = - 1.
H. BROCARD, P. F. TEILHET, "The equation
X2 - ay2 =- _2,"
L'Intermediare des mathematiciens, vol. X, p. 277, vol.
XII, p. 81, Paris, 1903, 1905.
R. W. D. CHRISTIE, A. CUNNINGHAM, R. F. DAVIs,
A. H. BELL, "Without any continued fraction procedure to
solve x2 - 149y2 = 1 and to generalize the method,"
Mathematical questions from the Educational Times,
vol. VI (2), p. 87, London, 1904. [Nearly the same in the
Educational Times, vol. LVI, p. 269, London, 1903, and
vol. LVII, p. 200, 275, London, 1904.]
Some of these methods remind us of Brahmagupta.
Conformal division is used. See A. Cunningham, "On
the connection of quadratic forms," Proceedings of the
London Mathematical Society, vol. XXVIII, p. 289,
London, 1896. Bell makes use of a combination of triple
series of arithmetical progressions. Christie uses primitive roots.
R. W. D. CHRISTIE, "Primitive roots applied to the
Pellian equation," Mathematical questions from the
Educational Times, vol. VI (2), p. 98, London, 1904.
R. W. D. CHRISTIE, "In the Pellian equation with any
prime of the form 4m + 1 and convergents, p,', qn, as
usual, prove qn2 + qn+2 = q2n++," Educational Times,
vol. LVII, p. 41, London, 1904.
See under Euler on p. 68. Example,
X2 - 277y2 = - 1,
then
(22,213)2 + (6,524)2 = 535,979,945,
where
(8,920,484,118)2- 277(535,979,945)2 = - 1.