The Pell equation, by Edward Everett Whitford.

126

THE PEIL EQUATION

we shall have (except for M = 1) for some values t, u',
less than t, u, either
Mt'2 - N'2 = 1; MN = A;
or
Mt,2 -_N 2 = 2, MN = A.
K. E. HOFFMAN, "Uber die Kettenbruchentwickelung
fur die Irrationale 2. Grades," Archiv der Mathematik
und Physik, vol. LXIV, p. 1, Leipzig, 1879.
The author applies continued fractions to the Pell
equation.
A. KUNERTH, "Praktische Methode zur numerischen
Auflosung unbestimmter quadratischer Gleichungen in
rationalen Zahlen," Situngsberichte der kaiserlichen
Akademie der Wissenschaften, mathematisch-naturwissenschaftliche Classe, vol. LXXVIII (Abtheilung 2), p.
327, Vienna, 1879.
S. ROBERTS, C. LEUDESDORF, EVANS, "Show that the
triangular numbers which are also squares are given by
J (I + -2)2 _ (1 - +2 )2m 2
4 ~2
Mathematical questions from the Educational Times,
vol. XXX, p. 37, London, 1879.
By decomposing the expression we can obtain remarkable laws of derivation of the successive squares,
12.12, (1 + 1)2(2.1 + 1)2, (2.2 + 3)2(2 + 3)2
1, 22.32, 72.52,
(7 + 5)2(2.5 + 7)2, (2.12 + 17)2(12+17)2,
122.172, 412.292, *..
EVANS, J. W. SHARPE, NASH, "If pn/qn be the last
convergent in the first period of Al expanded as a continued fraction, and r the greatest integer contained in