The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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modern symbols the proof is as follows:
If
aq2 + s = p,
and
aql2 + s1 = pi2,
multiplying both members of the first equation by pl2,
and substituting aq12 + sl for its equal, p12, when it is
multiplied into s, we have
aq2p12 + saql2 + SS1 = p2l2.
In the second term of the left member substituting for s
its equal p2 - aq,2 we have
aq2p12 + aq2p - a2q2q12 + ss =p p2p2.
Transposing the negative term, and adding == 2appiqqi
to both sides, we have
aq2p12 ' 2applqql + aq12p2 + ssi
= p2pl2 = 2applqqi + a2q2q22,
and
a(qpl = qip)2 + SSI = (ppi — 1 aqql)2.
Then
x = ppi = aqql, y = qpl I qlp
are solutions of the equation
ay2 + ss1 = X2.
By repeating these steps and by applying the last part
of the rule ssl becomes equal to 1; that is,
ay2 + 1 = X2
is solved. Then, as Bhaskara remarks, "by virtue of
[a variety of] assumptions and by composition either by
sum or difference, an infinity of roots may be found."
Let us examine "the splendor point of the collected
wisdom of the Hindus," the method of solution which
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