The Pell equation, by Edward Everett Whitford.

76

THE PELL EQUATION

no solution when B equals 5.41. If r2 - Bs2 = - 1,
nu must be odd; and this is only possible if u is also odd.
In case u is an even number this equation is never solvable
in integers. If, however, u is odd the equation can be
solved by the formulas just given, by substituting an odd
number for n.
In his Additions1 to Euler's Algebra Lagrange also gave
complete solutions of the equations
2 - By2 = 1
and
x2 - By2 = E.
The two indeterminate equations x2 - Ay2 = 1 and
x2- Ay2 = 4 are of primary importance in the theory
of quadratic forms of a positive and non-square determinant. When the complete solution of these equations
is known we can deduce from the single representation
of a number by a form every representation of the same
set, and from a single transformation of either of two
equivalent forms into the other, every similar transformation. Gauss has transformed the problem of the Pell
equation by his method of substitutions. He has avoided
the use of continued fractions and has shown that if we
form by the method he indicates the period of a quadratic
form of determinant A we may infer the complete solution
of the equations x2 - Ay2 = 1 and x2 - Ay2 = 4 from
the automorphics of any reduced form, according as the
form is properly or improperly primitive.2 The following
theorem from the Disquisitiones Arithmeticae3 shows how
he connected the Pell equation with the theory of quadratic forms.
1 J. L. Lagrange, "Euler's algebra with additions by M. de la Grange,"
translated by Hewlett, Chap. VIII, p. 578, London, 1840. On this page,
through a misunderstanding of Ozanam, Lagrange seems to think that
the solution of Lord Brouncker should be credited to Fermat.
2 C. F. Gauss, "Disquisitiones arithmeticae," ~~ 198-202, p. 259-273,
Leipzig, 1801.
3 Op. cit., ~ 162. "Werke," vol. I, p. 129, 2d ed., G6ttingen, 1870.