The Pell equation, by Edward Everett Whitford.

THE PELL EQUATI()ON

23

Therefore
y=2
and
x= 11.
The equation1
Ay2 + C = x2
is handled by Diophantus as follows. It can be rationally
solved if (1) A is a positive square, say a2. Thus
a2y2 + C = X2
In this case x2 is put equal to (ay i m)2.
Therefore
a2y2 + C = (ay n m)2,
and
C - m2
y = ~ d 2ma
(m and the doubtful sign being always assumed so as
to give x a positive value.) When (2) C is positive and
a square number, say c2,
Ay2 + c2 = 2.
Here we may let
x = my M = c.
Therefore
Ay2 + c2 = (my = c)2,
and
2mc
y= A
Diophantus shows how when one solution to the
equation is known any number of others may be found.
The lemma to VI, 15,2 reads: "Given two numbers,
if, when some square is multiplied into one of the numbers
and the other number is subtracted from the product,
the result is a square, another square larger than the
1G. H. F. Nesselmann, "Die Algebra der Griechen," p. 329, Berlin,
1842.
2 T. L. Heath, "Diophantus," p. 238.