1911 Encyclopædia Britannica/Number/Automorphs

35. Automorphs. The Pellian Equation.—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.