35. Automorphs. The Pellian Equation.—A primitive form
is, by definition, equivalent to itself; but it may be so in more ways than one. In order that
may be transformed into itself by the substitution
, it is necessary and sufficient that
![{\displaystyle {\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}={\begin{pmatrix}{\tfrac {1}{2}}(t+bu),&-cu\\au,&{\tfrac {1}{2}}(t-bu)\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62731d5e75b128a3b09940e48c7d585daf4b1845)
where
is an integral solution of
.
If
is negative and
, the only solutions are
;
gives
;
gives
. On the other hand, if
the number of solutions is infinite, and if
is the solution for which
have their least positive values, all the other positive solutions may be found from
.
The substitutions by which
is transformed into itself are called its automorphs. In the case when
we have
, and
any solution of
.
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
into a periodic continued fraction.
The form
may be improperly equivalent to itself; in this case all its improper automorphs can be expressed in the form
![{\displaystyle {\begin{pmatrix}\lambda ,&(\kappa +b\lambda )/2a\\(\kappa -b\lambda )/2a,&-\lambda \end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c216fc7b5c01707ba7b4d31a193b530a5ab74ae3)
where
. In particular, if
the form
is improperly equivalent to itself. A form improperly equivalent to itself is said to be ambiguous.