1911 Encyclopædia Britannica/Number/Quadratic forms

32. Quadratic forms.&mdash;It will be observed that the solution of the linear congruence $$ax\equiv n \pmod{m}$$ leads to all the representations of $$b$$ in the form $$ax+my$$, where $$x, y$$ are integers. Many of the earliest researches in the theory of numbers deal with particular cases of the problem: given four numbers $$m, a, b, c$$, it is required to find all the integers $$x, y$$ (if there be any) which satisfy the equation $$ax^2+bxy+cy^2=m$$. Fermat, for instance, discovered that every positive prime of the form $$4n+1$$ is uniquely expressible as the sum of two squares. There is a corresponding arithmetical theory for forms of any degree and any number of variables; only those of linear forms and binary quadratics are in any sense complete, as the difficulty of the problem increases very rapidly with the increase of the degree of the form considered or of the number of variables contained in it.

The form $$ax^2+bxy+cy^2$$ will be denoted by $$(a, b, c)(x,y)^2$$ or more simply by $$(a, b, c)$$ when there is no need of specifying the variables. If $$k$$ is the greatest common factor of $$a, b, c$$, we may write $$(a, b, c) = k(a', b', c')$$ where $$(a', b', c')$$ is primitive form, that is, one for which $$\operatorname{dv}(a', b', c') = 1$$. The other form is then said to be derived from $$(a', b', c')$$ and to have a divisor $$k$$. For the present we shall concern ourselves only with primitive forms. Writing $$\mathrm{D}=b^2-4ac$$, the invariant $$\mathrm{D}$$ is called the determinant of $$(a, b,c)$$, and there is a first classification of forms into definite forms for which $$\mathrm{D}$$ is negative, and indefinite forms for which $$\mathrm{D}$$ is positive. The case $$\mathrm{D}=0$$ or a positive square is rejected, because in that case the form breaks up into the product of two linear factors. It will be observed that $$\mathrm{D}\equiv 0, 1 \pmod{4}$$ according as $$b$$ is even or odd; and that if $$k^2$$ is any odd square factor of $$\mathrm{D}$$ there will be forms of determinant $$\mathrm{D}$$ and divisor $$k$$.

If we write $$x' = \alpha x+ \beta y, y' = \gamma x+ \beta y$$, we have identically
 * $$(a, b, c)(x',y')^2=(a', b', c')(x,y)^2$$

where
 * $$a' = a\alpha^2+b\alpha\gamma+c\gamma^2$$
 * $$b' = 2a\alpha\beta+b(\alpha\delta+\beta\gamma)+2c\gamma\delta$$
 * $$c' = a\beta^2+b\beta\delta+c\delta^2$$

Hence also
 * $$\mathrm{D}' = b'^2-4a'c' = (\alpha\delta-\beta\gamma)^2(b^2-4ac)=(\alpha\delta-\beta\gamma)^2\mathrm{D}$$.

Supposing that $$\alpha, \beta, \gamma, \delta$$ are integers such that $$\alpha\delta-\beta\gamma = n$$, a number different from zero, $$(a, b, c)$$ is said to be transformed into $$(a', b', c')$$ by the substitution $$\begin{pmatrix} \alpha, & \beta \\ \gamma, & \delta \end{pmatrix}$$ of the $$n\mbox{th}$$ order. If $$n^2=1$$, the two forms are said to be equivalent, and the equivalence is said to be proper or improper according as $$n=1$$ or $$n=-1$$. In the case of equivalence, not only are $$x', y'$$ integers wherever $$x, y$$ are so, but conversely; hence every number representable by $$(a, b, c)$$ is representable by $$(a', b', c')$$ and conversely. For the present we shall deal with proper equivalence only and write $$f\sim f'$$ to indicate that the forms $$f, f'$$ are properly equivalent. Equivalent forms have the same divisor. A complete set of equivalent forms is said to form a class; classes of the same divisor are said to form an order, and of these the most important is the principal order, which consists of the primitive classes. It is a fundamental theorem that for a given determinant the number of classes is finite; this is proved by showing that every class must contain one at least of a certain finite number of so-called reduced forms, which can be found by definite rules of calculation.