The Pell equation, by Edward Everett Whitford.

134

THE PELL EQUATION

From the fundamental solution, x0, yo, of the equation
x2- (a2 + l)y2 = -N, all the solutions, x, y, of both
equations are given by the identity
x + a 1 ( + y = ( a + 1)(a + 4a2 + 1)
in which n runs through all odd values.
G. FRATTINI, "Dell' analisi indeterminata di secundo
grado," Periodico di matematica, vol. VI, p. 169, vol.
VII, p. 7, 49, 88, 119, 172, Rome, 1891, 1892.
C. A. ROBERTS, A. MARTIN, "A table of the square roots
of the prime numbers of the form 4n + 1 less than 10,000
expanded as periodic continued fractions," Mathematical
magazine, vol. II, p. 105, Washington, 1892.
Comment is made upon the fact that the number of
terms in the period of the continued fraction for 4N
never exceeds 2a where a2 is the largest square less than N,
provided N < 1,000; and it is asked whether this can
be proved generally. It can not. See the numbers
8,269, 8,941, 9,949, 4,909. I have found several numbers
smaller than these for which the period exceeds 2a, namely,
1,726, 1,831, 2,011. See the table in appendix A.
G. FRATTINI, "Due propositioni della teoria dei numeri
e loro interpretazione geometrica," Atti della Reale
Accademia dei Lincei, vol. I (5), p. 51, Rome, 1892.
The author gives several theorems for deducing integral
solutions of the equations
2 - Dy2 = N,
together with geometric interpretations.
G. FRATTINI, "A complemento di alcuni teoremi del sig.
Tchebicheff," Atti della Reale Accademia dei Lincei, vol.
I (5), p. 85, Rome, 1892.
This article discusses the solutions in positive integers
of the equations x2 - Dy2 = - N. The values of x in