The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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from which we get, since
30.2 + 1.60 = 120
and
3.1-30 + 60.2 = 210,
L 120, G 210, A 900.
The next rule is one which saves much labor in the
computation of the solutions of many of the equations,
but it does not produce the fundamental solution, nor
does it give integral values unless y is even. "Rule:
When the additive is four, the square of the last root,
less three, being halved and multiplied by the last, is a
last root; and the square of the last root, less one, being
divided by two and multiplied by the first, is a first root
of additive unity." If
X12 - Ay2 = 4,
then
(12 - 3) y(12- 1)
2 ' 2
are roots of the equation
2 - Ay2 = 1.
From the rule for "subtractive four," if
x12 - Ay12 - 4,
we find in a similar manner that
y = (X + 3) X12 +
are roots of the equation
x2 - Ay2 = - 1.
These rules of Brahmagupta and those of Bhaskara