The Pell equation, by Edward Everett Whitford.

22

THE PELL EQUATION

approximations,l with their connection indicated:

Surd. Heron's approximation.
450 7 +,
1 1 1
4135 l 11 — + -
2 14 21'
2 1
/1575 ~ 39 + 2 +
2 1
216 c 14 ++ 33
1
1 720 c\ 26 ++,
2 3 1

Equivalent
fraction.
99
14'
244
21'
2024
51 '
485
33 '
161
6 '

Corresponding to the
fundamental solution of
X2 - 50y2 = 1;
x2 - 135y2 = 1;
x2 - 1575y2 = 1;
x2 - 216y2 =1;
X2- 720y2 = 1.

Diophantus flourished about 250 A.D., or not much
later. In discussing his relation to the problem, we must
remember that he did not avoid fractional solutions to
indeterminate equations but sought merely rational
solutions. But there are many cases in which he actually
does find integral solutions. In Book V of his Arithmetica,3 problem 9, we have the equation

whence
Therefore

26y2 + 1 = x2;
26y2 + 1 = (5y + 1)2, say.
y= 10,

where the corresponding value of x would be 51. In
Book V, 11, we have the equation

whence

30y2 + 1 = x2;
30y2 + 1 = (5y + 1)2, say.

1 S. Giinther, loc. cit.
2 This is as old as Pythagoras. T. L. Heath, "The works of Archimedes,"
p. lxxix, Cambridge, 1897.
3 T. L. Heath, "Diophantus of Alexandria, a study in the history of
Greek algebra," 2d ed., p. 207, Cambridge, 1910.