The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

35

are as follows: Choose an equation
(2) aq2 + s = p2
where p, q, are relatively prime, and where to abbreviate
computation, s is made as small as possible, although
this latter assumption is not essential. Then p, q, s, are
known. Two integers, q1 and ri, are now determined such
that
p +q r1
(3) q 1
which can be done by selecting an appropriate value for
rl. For this purpose the Hindus used a process called
pulverizer, analogous to that of continued fractions. At
the same time r12 - a is made as small as possible. Then
the expresssion
ri2 - a
81 -
is an integer.1 From this we obtain
aql2 + S1 = p2.
1To prove that (r12 - a)/s is always an integer substitute the value
of rl from the equation,
p + qrl
thus:
2q12 - 2sqlp + p2 _
r2 - a _-q2_ s2q,2 - 2sqlp + p2 - q2a.
s s sq2
Substituting for p2 - q2a its value s, from the second equation, we have
r2 - a s2q2 - 2sqlp+ s sq12-2qp +1 sql -p2q - 2 + p2ql2 - 2qlp + 1
S sq2 q2 q2
Substituting for s - p2 its value - aq2, from (2), we have
rl2 -a -aq2q2 + (pql - 1)2 _aq2 + (p — 12
s q q
Now combining (2) and (3) to eliminate s, and
qip2 - aq2q1 = p + qrl,
and
p(pql - 1) = q(rl + aqqi);