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
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
where . In particular, if the form is improperly equivalent to itself. A form improperly equivalent to itself is said to be ambiguous.