The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

27

tion. In the eighteenth chapter of his algebra, which he
called Brahme-sphuta-sidd'hanta, we have the following
"Rule:l A root [is set down] two fold: and [another deduced] from the assumed square multiplied by the multiplier, and increased or diminished by a quantity assumed.
The product of the first [pair] taken into the multiplier,
with the product of the last added, is a 'last' root. The
sum of the products of oblique multiplication is a 'first'
root. The additive is the product of the like additive
or subtractive quantities. The roots [so found], divided
by the [original] additive or subtractive quantity, are
[roots answering] for additive unity." That this rule
applies to the solution of the Pell equation will be more
evident by the statement of the problem following the
rule.
"Question 27. Making the square of the residue of
signs and minutes on a Wednesday, multiplied by ninetytwo... with one added to the product [afford]... an
exact square, [a person solving this problem] within a
year [is] a mathematician."
In modern language solve the indeterminate equation
X2 - 92y2 = 1.
Brahmagupta's procedure is evident from a careful study
of the above rule. In simplifying it we let L stand for
"least," which corresponds to y in the equation and to
"first" in the rule, and we let G stand for "greatest,"
which corresponds to x in the equation and to "last" in
the rule. Brahmagupta assumes that the square is 1,
its root is "least" root and he sets it down twice, thus:
L 1,
L 1.
He multiplies this square by 92 and adds 8 to make it
1 H. T. Colebrooke, "Algebra, with arithmetic and mensuration from
the Sanscrit of Brahmegupta and Bhaskara, " p. 363, London, 1817.