The Pell equation, by Edward Everett Whitford.

26

THE PELL EQUATION

Then when
u= 1, v= 1,
satisfy the equation
x2 _ 3y2 = 2,
we have
(u2 + 3v2)p - 6uvq 4p + 6q
Pi = u2v2 -2 - (2p + 3q),
and
2uvp + (u2 + 3v2)q 2p +- 4q
q = 32 = - (p + 2q),
and the positive values for pi, ql, could also be taken since
they satisfy the equation
pl2 - 3ql2 = 1.
Thus in order to get a general solution, Diophantus
required two known solutions of the original equation,
or one of the original and one of an auxiliary equation.
Although the solution of the Pell equation is not
explicitly found in the writings of Aryabhatta (c. 525
A.D.), it is by no means certain that he knew nothing of
it,l for he recorded only so much algebra as he conceived
to be necessary for his astronomy. Moreover, Aryabhatta had a definite idea of the approximation formula
aa2 + r a + 2
which would lead to solutions of the equation, for he
says in one of his rules:2 "Square having been subtracted
from square always the non-square must be divided by
double the square root. The quotient in a place set
apart is the root."
Turning to the works of Brahmagupta (c. 650 A.D.),
we find a wealth of material bearing directly on the equa1 H. Hankel, p. 203.
2 G. R. Kaye, " Notes on Indian mathematics," Journal and Proceedings
of the Asiatic Society of Bengal, vol. IV, No. 3, p. 119, Calcutta, 1908.
At the conclusion of his article the author gives a bibliography of Indian
mathematics.