1911 Encyclopædia Britannica/Number/Problem of Representation

34. Problem of Representation.&mdash;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.