The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

115

G. L. DIRICHLET, "Recherches sur diverses applications de l'analyse infinitesimale a la theorie des nombres,"
Journal ffir die reine und angewandte Mathematik, vol.
XIX, p. 324, vol. XXI, p. 1, Berlin, 1839, 1840.
C. D'ANDREA, "Trattato elementare di aritmetica e
d'algebra," vol. II, p. 671, Naples, 1840.
That t2 - Au2 = 1 is always solvable in integers is
proved through continued fractions.
C. G. J. JACOBI, "Uber die Kreistheilung und ihre
Anwendung auf die Zahlentheorie," Journal fur die reine
und angewandte Mathematik, vol. XXX, p. 166, Berlin,
1846.
If p is a prime of the form 4n + 1 and a designates the
quadratic residues between 0 and ~p, then
P-l ar
(x + y l-p) /p = 22 I sin2 -
where x2 - py2 = - 4, and II designates the product of
the collected factors sin2 air/p; and if q is a prime of the
form of 8n + 3
x+yq =. 2.nsin( + )
where x2 - qy2 = - 2. There is a table of the separations of the primes p of the form 4n + 1 into the sum of
two squares up to p = 11,981. This article was first
published in Monatsberichte der K6niglichen Akademie
der Wissenschaften, 1837.
F. ARNDT, "Bemerkungen iber die Verwandlung der
irrationalen Quadratwurzel in einen Kettenbruch," Journal fur die reine und angewandte Mathematik, vol. XXXI,
p. 343, Berlin, 1846.
The author discusses p2 - Aq2 = 1, when A is an odd
number, when A is an odd power of 2, and when A = 2A'
where A' is odd. The work of Legendre in his "Theorie
des nombres" is generalized and extended.