Elements of Algebra

388 ELEMENTS PART II. which gives $hpq - hgy = frs - fky$; consequently, $y = \frac{frs-hpq}{fk-hg}$, and $x=\frac{kpq-grs}{fk-hg}$*; but if we choose to have $x$ and $y$ expressed in integer numbers, we must give such values to the letters, $p$, $q$, $r$, and $s$, that the numerator may be really divisible by the denominator; which happens either when $p$ and $r$, or $q$ and $s$, are divisible by that denominator. 165. To render this more clear, let there be given the formula $x^2 - y^2$, composed of the factors $(x+y) \times (x-y)$. Now, if this must be resolved into a greater number of factors, we may make $x+y=pq$, and $x-y=rs$; we shall then have $x = \frac{pq+rs}{2}$, and $y = \frac{pq-rs}{2}$; but, in order that these values may become integer numbers, the two products, $pq$ and $rs$, must be either both even, or both odd. For example, let $p=7$, $q=5$, $r=3$, and $s=1$, we shall have $pq=35$, and $rs=3$; therefore, $x=19$, and $y=16$; and thence $x^2 - y^2 = 105$, which is composed of the factors $7 \times 5 \times 3 \times 1$; so that this case is attended with no difficulty. 166. The second is attended with still less; namely, that in which the formula, containing two equal factors, may be represented thus: $(fx+gy)^2$, that is, by a square, which can have no other factors than those which arise from the root $fx+gy$; for if we make $fx+gy=pqr$, the formula becomes $p^2q^2r^2$, and may consequently have as many factors as we choose. We must farther remark, that one only of the two numbers $x$ and $y$ is determined, and the other may be taken at pleasure; for $x = \frac{pqr-gy}{f}$; and it is easy to give $y$ such a value as will remove the fraction. The easiest formula to manage of this kind, is $x^2$; if we make $x=pqr$, the square $x^2$ will contain three square factors, namely $p^2$, $q^2$, and $r^2$. 167. Several difficulties occur in considering the third case, which is that in which our formula cannot be resolved * For, since $fx+gy=pq$, and $hx+ky=rs$, we have $y = \frac{pq-fx}{g}$, and $y = \frac{rs-hx}{k}$; then $\frac{pq-fx}{g} = \frac{rs-hx}{k}$; whence, $fkx-hgx=kpq-grs$, and, consequently, $x = \frac{kpq-grs}{fk-hg}$.