1911 Encyclopædia Britannica/Number/Automorphs

35. ''Automorphs. The Pellian Equation.''&mdash;A primitive form $$(a, b, c)$$ is, by definition, equivalent to itself; but it may be so in more ways than one. In order that $$(a, b, c)$$ may be transformed into itself by the substitution $$\begin{pmatrix} \alpha, & \beta \\ \gamma, & \delta \end{pmatrix}$$, it is necessary and sufficient that
 * $$\begin{pmatrix} \alpha, & \beta \\ \gamma, & \delta \end{pmatrix} = \begin{pmatrix} \tfrac{1}{2}(t+bu), & -cu \\ au, & \tfrac{1}{2}(t-bu) \end{pmatrix}$$

where $$(t, u)$$ is an integral solution of
 * $$t^2-\mathrm{D}u^2=4$$.

If $$\mathrm{D}$$ is negative and $$-\mathrm{D}>4$$, the only solutions are $$t=\pm 2, u=0$$; $$\mathrm{D} = -3$$ gives $$(\pm 2, 0), (\pm 1, \pm 1)$$; $$\mathrm{D}= -4$$ gives $$(\pm 2, 0), (0, \pm 1)$$. On the other hand, if $$\mathrm{D}>0$$ the number of solutions is infinite, and if $$(t_1, u_1)$$ is the solution for which $$t, u$$ have their least positive values, all the other positive solutions may be found from
 * $$\frac{t_n+u_n\sqrt{}\mathrm{D}}{2}=\left(\frac{t_1+u_1\sqrt{}\mathrm{D}}{2}\right)^n (n=2, 3, 4 \dots)$$.

The substitutions by which $$(a, b, c)$$ is transformed into itself are called its automorphs. In the case when $$\mathrm{D} \equiv 0 \pmod{4}$$ we have $$t = 2\mathrm{T}, u = 2\mathrm{U}, \mathrm{D} =4\mathrm{N}$$, and $$(\mathrm{T}, \mathrm{U})$$ any solution of
 * $$\mathrm{T}^2-\mathrm{N}\mathrm{U}^2 = 1$$.

This is usually called the Pellian equation, though it should properly be associated with Fermat, who first perceived its importance. The minimum solution can be found by converting $$\sqrt{}\mathrm{N}$$ into a periodic continued fraction.

The form $$(a, b, c)$$ may be improperly equivalent to itself; in this case all its improper automorphs can be expressed in the form
 * $$\begin{pmatrix} \lambda, & (\kappa+b\lambda)/2a \\ (\kappa-b\lambda)/2a, & -\lambda \end{pmatrix}$$

where $$\kappa^2-\mathrm{D}\lambda^2 = 4ac$$. In particular, if $$b \equiv 0 \pmod{a}$$ the form $$(a, b, c)$$ is improperly equivalent to itself. A form improperly equivalent to itself is said to be ambiguous.