The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

141

The author compares his method for the study of
quadratic forms with that of Gauss and Dirichlet and
shows the superiority of his own method, for example in
the solution of the equation
(a2 - 4)y - 4x2 = 1.
E. DE JONQUIERES, "Formules generales donnant des
valeurs de D pur lesquelles l'6quation t2 - Du2 = - 1
est resoluble en nombres entiers," Comptes rendus de
l'Academie, vol. CXXVI, p. 1837, Paris, 1898.
Two of the theorems proved are: the equation is
solvable when D is of the form 4n2 + n + 5; when
D = a2(n2 + 1) and n is a multiple of a, the equation is
impossible.
G. WERTHEIM, "Uber die Ausziehung der Quadrat- und
Kubikwurzeln bei Heron von Alexandria," Zeitschrift fiir
mathematischen und naturwissenschaftlichen Unterricht,
vol. XXX, p. 253, Leipzig, 1899.
A. PALMSTROM, "Les solutions ordinaires des equations
x2 + 1 = 2y2 forment elles le systeme complet des solutions?" L'Intermediare des mathematiciens, vol. VI,
p. 40, Paris, 1899.
Puisse, op cit., vol. V, p. 31, 1898, asks a question
implied in the above. See J. A. Serret, "Cours d'algebre
superieure," p. 77, 5th ed., Paris, 1885, and P. Bachman,
"Die Elemente der Zahlentheorie," p. 182, Paris, 1892.
A. PALMSTROM, "Proprietes relatives a l'6quation
x2 - Ay2 = 1," L'Intermediare des mathematiciens, vol.
VI, p. 210, Paris, 1899.
The properties discussed are
X2n + Y2n A = (X1 + Y1 IA)2n = (Xn + Yn A)2,
X2n = Xn2 + AYn2 = 2Xn2 - 1, and Y2n = 2XYn.
G. WERTHEIM, "Pierre Fermats Streit mit John Wallis,
ein Beitrag zur Geschichte der Zahlentheorie," Abhand