The Pell equation, by Edward Everett Whitford.

40

THE PELL EQUATION

It is true that the approximation formulas for square
root used by Alkarkhi and others starting with
/a2 +r a+ 2a+
do not show any connection with the solutions of the
simplest Pell equations but these were not the only
approximations used by the Arabs. El-Hassar,1 in his
extraction of an irrational root, says: "When it is
asked what is the square root of 5, take the nearest square
number in 5, this equals 4, subtract it from 5, the remainder
is 1, divide this by 4, this gives 1/4, and add this to the
root of 4, which equals 2, this gives 2}, and this is the
approximate root of 5." This is the usual approximation a/a2 + r c a + r/2a, and 2a + r, 2a, is a solution
of the Pell equation
x2 - Ay2 = r2.
In particular, the approximation obtained here, 9/4,
gives
x = 9, y = 4,
as a solution of
2 - 5y2 = 1.
El-Hassar then obtains a still closer approximation,
217, virtually by the use of the formula, -— r \2
+a2+r c a+r - 2
2 a+)2a
thus giving another solution of the equation
2 - 5y2 = 1,
namely
x = 161, y = 72.
The solution of
X2 - Ay2 = r4,
1H. Suter, "Das Rechenbuch des Abu Zakarija el-Hassar," Bibliotheca mathematica, vol. II (3), p. 37, Leipzig, 1901.