The Pell equation, by Edward Everett Whitford.

28

THE PELL EQUATION

yield a square, the root of which, 10, he calls "greatest"
and sets down the result as follows:
L 1, G 10, A 8,
L 1, G 10, A 8,
where A 8 denotes the number which must be added to
92L2 to make the sum a perfect square. Then by the
rule, 12.92 + 102 makes a new G, and 1.10 + 1-10
makes a new L, where the new number to be added is
8.8, giving
L 20, G 192, A 64.
By the last part of the rule, dividing by 8, we obtain,
5 &L
L2, G 3 A 1.
If L and G are divided by 8, A is divided by 8 8. Repeating the rule on
5
L, - <G % Al, 1,
L2, G A1,
we get
L 120, G 1151, A 1,
whence
y = 120, x= 1,151
is the fundamental solution of
x2 92y2 = 1.
The next rule shows that the Hindus knew how from
one solution to obtain others. "Rule: Putting severally
for additive unity under roots for the given additive or
subtractive, the 'last' and 'first' roots [thence deduced
by composition] serve for the given additive or subtractive."
Example.
L 30, G 60, A 900,
L 1, G 2, A 1,