The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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Along the line suggested by the work of Gauss, Dirichlet1
developed the following theorem which shows the intimate
connection of the equation with the theory of quadratic
forms:
If (a, b, c) is a form2 of determinant D and divisor o-,
and if
(k)
is any proper substitution through which (a, b, c) is transformed into itself, then always
t - bu - cu au t + bu
X = -, I =, v=-, p= = -,
where t, u, are integers which satisfy the indeterminate
equation t2 - Du2 = o-2; and conversely from every solution, t, u, of the equation, the above formulas afford a
substitution through which the form (a, b, c) is transformed into itself.
From the theory of quadratic forms,3 we have the
following equations:
(1) Xp - Jv =- 1,
(2) a2 + 2bXv + cv2 = a,
(3) aXgy + b(Xp + pv) + cvp = b,
(4) from (1) Xp = Av + 1,
1 G. L. Dirichlet, "Vorlesungen ilber Zahlentheorie," 4th ed. (edited by
Dedekind), p. 149, p. 201, Braunschweig, 1894. See also Dirichlet,
"Recherches sur les forms quadratiques a coefficients et a indeterminees
complexes," Werke, vol. I, p. 535-618, Berlin, 1889; Journal fiur die reine
und angewandte Mathematik, vol. XXIV, p. 291, Berlin, 1842; and
"Recherches sur diverses applications de l'analyse infinitesimale a la
th6orie des nombres," Journal fur die reine und angewandte Mathematik,
vol. XIX, p. 324, and vol. XXI, p. 1, Berlin, 1839, 1840.
2 Gauss and Dirichlet designate ax2 + 2bxy + cy2 by (a, b, c), but
Kronecker designates ax2 + bxy + cy2 by the same symbol (a, b, c). The
greatest common divisor of a, 2b, c is designated by o, and D = b2 - ac.
3 G. L. Dirichlet, "Zahlentheorie," 4th ed., ~ 54, p. 130, Braunschweig,
1894.
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