The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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and says that they have been worked out independently
by G. W. Tenner, and many writers ascribe them to the
latter.
It is now shown that
ifx = 1, y = 0, then.x2 = zy2 + 1;
if x = (v), y = 1, then x2 = zy2 - a;
if x = (v, a), y = (a), then x2 = zy2 -+;
if x = (v, a, b), y = (a, b), then x2 = zy2 - 7;
if x = (v, a, b, c), y = (a, b, c), then x2 = zy2 + 5;
if x = (v, a, b, c, d), y = (a, b, c, d), thenx2 =zy2 - e;
If one of the letters 3, 5,. = 1, we have a solution of
the equation
x2 - y2 = 1.
But none of these letters can equal =- 1 unless at the
same time the corresponding index is 2v. If then any
period contains the index 2v and we place x, y, equal to
the convergent values which correspond to this period,
we obtain x2 - zy2 = 1, if the number of indices in the
period is even; and x2 - zy2 = - 1, if the number is
odd. In the first case we have directly the solution
sought; in the other case by taking two periods' together,
we get a period having an even number of closing mem1There is an error on this point in M. Cantor, "Vorlesungen fiber
Geschichte der Mathematik," vol. IV, p. 159, 3d ed., Leipzig, 1908; for
there it says two periods further and convergents at the end of the third
period. By combining two periods with an odd number of indices a period
is made having an even number of indices and closing with 2v, which would
therefore furnish a solution of the Pell equation x2 - zy2 = 1. But if as
the text directs, we proceed two periods further than the close of the first
odd period we are at the close of another odd period and do not have a
solution of the equation x2 - zy2 = 1 but of the equation x2 - zy2 = - 1.
To get the solution of x2 - zy2 = 1 we must not take the convergent at the
close of the third period, as the text says, but at the end of the second or
fourth or some even number of periods. If we took the convergent at the
end of the third period we would only have the solution of x2 - zy2 = - 1.