1911 Encyclopædia Britannica/Number/Problem of Representation

34. Problem of Representation.—It is required to find out whether a given number $m'$ can be represented by the given form $(a',b',c')$. One condition is clearly that the divisor of the form must be a factor of $m'$. Suppose this is the case; and let $m,(a,b,c)$ be the quotients of $m'$ and $(a',b',c')$ by the divisor in question. Then we have now to discover whether $m$ can be represented by the primitive form $(a,b,c)$. First of all we will consider proper representations

$m=(a,b,c)(\alpha ,\gamma )^{2}$

where $\alpha ,\gamma$ are co-primes. Determine integers $\beta ,\delta$ such that $\alpha \delta -\beta \gamma =l$, and apply to $(a,b,c)$ the substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$; the new form will be $(m,n,l)$, where

$n^{2}-4ml=\mathrm {D} =b^{2}-4ac$.

Consequently $n^{2}=\mathrm {D} {\pmod {4m}}$, and $\mathrm {D}$ must be a quadratic residue of $m$. Unless this condition is satisfied, there is no proper representation of $m$ by any form of determinant $\mathrm {D}$. Suppose, however, that $n^{2}=\mathrm {D} {\pmod {4m}}$ is soluble and that $n_{1},n_{2},\!\!\And \!\!\!\!\mathrm {c} .$ are its roots. Taking any one of these, say $n_{i}$, we can find out whether $(m,n_{i},l_{i})$ and $(a,b,c)$ are equivalent; if they are, there is a substitution ${\begin{pmatrix}\alpha ,&\beta \\\gamma ,&\delta \end{pmatrix}}$ which converts the latter into the former, and then $m=a\alpha ^{2}+b\alpha \gamma +c\gamma ^{2}$. As to derived representations, if $m=(a,b,c)(tx,ty)^{2}$, then $m$ must have the square factor $t^{2}$, and $m/t^{2}=(a,b,c)(x,y)^{2}$; hence everything may be made to depend on proper representation by primitive forms.