The Pell equation, by Edward Everett Whitford.

72

THE PELL EQUATION

has done with that of the first degree. We must believe
that he has applied himself to this research, for the
problem which he proposed to M. Wallis and all the
English geometers, is the keystone of the general solution of these equations. The problem is a particular
case of equations of the second degree in two unknowns,
and consists in finding two integral squares of which the
one being multiplied by any given non-square integer
and then subtracted from the other the remainder
should be equal to unity. Whether M. Fermat has not
continued his researches in this matter, or a record of
them has not come to us, it is certain that no traces of
them have been found in his works. It appears also
that the English geometers who have solved the problem
of M. de Fermat have not known all its importance for
the general solution of indeterminate problems of the
second degree. At least, we do not see that they have
ever made use of it and Euler is, if I do not mistake, the
first who has shown how by aid of this problem we are
able to find an infinity of solutions in whole numbers
of all the equations of the second degree in two unknowns
of which we already know one solution. Since the work
of M. Bachet (1613) up to the present time, with the
exception of the memoir which I gave the past year
upon the solution of indeterminate problems of the
second degree, the theory of this sort of problems has
properly speaking not been pushed beyond the first
degree."
The first admissible proof of the solvability of the
equation x2 - Ay2 = 1, as has been stated, was given by
Lagrange.1 He shows that in the development of 4A
we shall obtain an infinite number of solutions of certain
equations of the form x2 - Ay2 = B, and that by multiplying a sufficient number of these equations together,
member by member, we can deduce the solution of
x2 - Ay2 =1. Lagrange was dissatisfied with his firstJ. L. Lagrange, Miscellanea Taurinensia, loc. cit.