The Pell equation, by Edward Everett Whitford.

70

THE PELL EQUATION

bers with 2v. Then the values of x and y corresponding
to this second index 2v are taken. Or, if from the equation
x2 - zy2 = - I we take xi = 2x2 + 1, yi = 2xy,we have a
solution, xi, yl, of the equation xi2 - zy12 = 1.
These procedures lead in a convenient manner to the
solution of the equation x2 - zy2 = 1 in least integers.
Care should be taken not to overlook the solution of
x2 - zy2 = -1 in least integers in the particular case
in which z = a2 + 1, in which case the equation is satisfied by x = a, y = 1.
As already stated, Euler does not prove that we will
ever arrive at the index 2v, and unless this is shown we
are not sure that we will reach any solution beyond
x = 1, y = 0. It was left to Lagrange to clear this
matter up. H. J. S. Smith,1 one of the greatest English
authorities on the theory of numbers, says: "Euler
observed that x/y is necessarily a convergent to the value
of A/z, so that to obtain the numbers x and y it suffices to
develop i/z in a continued fraction. It is suggestive,
however, that it never seems to have occurred to him
that to complete the theory of the problem, it was necessary to demonstrate that the equation is always resoluble,
and that all its solutions are given by the development
of the a/z. His memoir2 contains all the elements necessary to the demonstration, but here, as in some other
instances, Euler is satisfied with an induction which
does not amount to a rigorous proof."
The following opinions may help us to estimate properly
the work of Fermat, Brouncker, and Euler, and they
form an appropriate introduction to the important
additions of Lagrange.
Legendre3 says: "Fermat is the first who seems to
1 H. J. S. Smith, "Report on the theory of numbers," British Association
report, p. 135, London, 1861; "Collected works," vol. I, p. 194, Oxford,
1894.
2 L. Euler, "Commentari arithmetical" vol. I, p. 316, St. Petersburg,
1849.
3 A. M. Legendre, "Theorie des nombres," 3d ed., ~ 36, Paris, 1830.