The Pell equation, by Edward Everett Whitford.

74 THE PELL EQUATION
where I E < I 'B. The procedure in outline is as follows:l Letting E = 1, we seek first a positive integer E
such that
/B - 1 < e < /B,
and for which B - e2 is divisible by 1. This goes without
saying, of course, but this is only a special case of B - e2
divisible by E where E may be different from 1. Therefore in this case we must take for c the integer immediately
below iB. As we now know E and E, we may obtain
the series, E, E1, E2, -"; el, E2,,; Xi, X2, Xs, " *;
by the aid of the following formulas,2
EE1 = B - e2,
EIE2 = B - E2,
E2E3 = B - E22,
E3E4 = B - e32,
(e1 = X1E1 - e,
e2 = X2E2 - 1,
e3 = X3E3 - 2,
> X1 > > 1,
B+e E1 IB+ E1
B_ - +l >,2 > B - _ 1,
E2 E2
JB -+ E2 > X3 > B -+ 2_ 1,
E3 E3
We continue in this construction until two members, E,
and E,+l are found which are identical with E and E1.
The proof that this always occurs is given at length.3
1 J. L. Lagrange, "Uber die L6sung..," loc. cit.
2 Op. cit., p. 48, p. 52.
3 Op. cit., p. 55.