The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

121

The first part of this problem is said to have been solved
by a student of Professor C. Gill at St. Paul's College,
Flushing, about 1842, and at that time the numbers were
the largest of the kind that had been found. x has 57
figures.
O. SCHL6MILCH, "Uber die Kettenbruchentwickelungen
fir Quadratwurzeln," Zeitschrift fir Mathematik und
Physik, vol. XVII, p. 70, Leipzig, 1872.
F. DIDON, C. MOREAU, "Solution of a question,"
Nouvelles annales de mathematiques, vol. XI (2), p. 48,
vol. XII (2), p. 330, Paris, 1872, 1873.
The indeterminate equation t2 - Du2 = 4, in which
D is of the form (4n + 2)2 + 1, n designating any positive
integer, 1, 2, 3,..., has no solution formed of two odd
numbers, and the solution which is made of the two smallest positive integers is
t = 16(2n + 1)2 + 2, u = 8(2n + 1).
L. MATTHIESSEN, "Allgemeine Auflisung der Gleichung
ax2 -* 1 = y2 in ganzen Zahlen," Zeitschrift fir Mathematik und Physik, vol. XVIII, p. 426, Leipzig, 1873,
B. MINNIGERODE, "Uber eine neue Methode die
Pell'sche Gleichung aufzul6sen," Nachrichten von der
K6nigliche Gesellschaft der Wissenschaften, No. 23, p.
619, Gottingen, 1873.
Reduced forms and continued fractions are used.
G. H. HOPKINS, HART, "If the sum of the squares of
two consecutive integers be equal to the square of another
integer, find their integral values, and show how to find
any number of particular solutions," Mathematical questions from the Educational Times, vol. XX, p. 63, London,
1874.
Dr. Hart gives as the general solution of

p2 - Nq2 = = 1, p = 2mr + r', q = 2ms + s'