The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

67

we have
B = a -A - v, > 1, b6 2v,
and
C = b - B v, y 1, c < 2v,
etc.,
the integers
A, B, C, D,... < v,
the integers,and the indices,, 1,
and the indices
a, b, c, d,... = 2v.
Euler declares that after the index 2v is reached, the values
a, b, c, * * repeat themselves and the whole development
begins anew, but he gives no proof that the index 2v exists.
In examples he shows that a, b, c, *.. and a, f3, y,.. are
repeated periodically, and for seven or eight different forms
of z like those mentioned on page 61 general values for the
indices and the Greek letters are given. The successive
convergents to the continued fractions are then investigated. They are developed according to the law apparent
in the following series,
v a b c *. m n *
1 v av+ 1 (ab + l)v + b M N nN+M
' 1' a ' ab + 1 ' ' P' Q' nQ+P'
There is a shorter algorism introduced for the convergents
as follows:
1 (v) (v, a) (v, a, b) (v, a, b, c) (v, a, b, c, d)
O' 1' (a) ' (a, b) ' (a, b, c) ' (a, b, c, d)
where
(v, a) = a(v) + 1; (v, a, b) = b(v, a) + (v);
(v, a, b, c) = c(v, a, b) + (v, a); *. (a) = al1 + 0;
(a, b) = b(a) + 1; (a, b, c) = c(a, b) + (a); *-*,

Euler states that he had proved the following transforma