The Pell equation, by Edward Everett Whitford.

98

THE PELL EQUATION

The present paper extends these solutions from the point
where the British committee stops to A = 1,700 inclusive.
It has been thought best not to omit the solution of the
important equation x2 - Ay2 = 1 even in the case in
which x2 - Ay2 = - 1 has a solution. In this respect the
table presents an added convenience, different from former
tables. The solutions have been made by first reducing
A1I to a continued fraction, and then constructing tables
for each number, as is shown in the two models given below
which not only give the solution, x, y, of the equation
x2 - Ay2 = 1 but also the solution, x, y, of the equation
x2 - Ay2 = B for the numbers | B I < 2 iA, when a
solution exists. In the first column are the indices of the
development of 1A; in the second and third columns the
values of x and y which substituted in the form x2 - Ay2
give the value named in the fourth column. In the first
line x is always taken as 1, and y as 0; and in the second
line x = the square root of the greatest square in A, and
y = 1. Thereafter any value of x is obtained by multiplying the value of x on the line above by the index
on the same line in the first column and adding to the
product the value of x in the next line above. The same
procedure gives the values of y. The fact that the middle
of the period is reached in the development of the continued fraction is indicated by parentheses around the
index and also, in the continued fraction table, around
the number beneath it which denotes the corresponding
value of the form x2 - Ay2. The two model tables differ
in that in the second the period of the continued fraction
contains an odd number of terms, and parentheses are
placed around the two indices in the center to indicate
this. That the number of terms in the period of -A is
odd is a necessary and sufficient condition for the solution of
x2 - Ay2 = - 1.
In such cases both the solution of
x2 - Ay2 = - 1