Elements of Algebra

CHAP. XIV. OF ALGEBRA. 433 $2ry + r^2 = \frac{q^2+3a}{2q} y + a$, or $4qry + 2qr^2 = (q^2 + 3a)y + 2aq$, or $2qr^2 - 2aq = (q^2 + 3a)y - 4qry$, and $y = \frac{2qr^2-2aq}{q^2+3a-4qr}$, where $q$ and $r$ may have any values, provided $x$ and $y$ become whole numbers; for since $p=y+r$, the numbers, $z$ and $v$, will likewise be integers. The whole depends therefore chiefly on the nature of the number $a$, and it is true that the condition which requires integer numbers might cause some difficulties; but it must be remarked, that the solution is already much restricted on the other side, because we have given the letters, $z$ and $v$, the values $x+y \pm 2p$, notwithstanding they might evidently have a great number of other values. The following observations, however, on this question, may be useful also in other cases. 1. When $xy+a$ must be a square, or $xy = p^2 - a$, the numbers $x$ and $y$ must always have the form $r^2 - as^2$ (Art. 176); if, therefore, we suppose $x = b^2 - ac^2$, and $y = d^2 - ae^2$, we find $xy = (bd - ace)^2 - a(be - cd)^2$. If $be - cd = \pm 1$, we shall have $xy = (bd - ace)^2 - a$, and, consequently, $xy + a = (bd - ace)^2$. 2. If we farther suppose $z = f^2 - ag^2$, and give such values to $f$ and $g$, that $bg - cf = \pm 1$, and also $dg - ef = \pm 1$, the formulæ $xz+a$, and $yz+a$, will likewise become squares. So that the whole consists in giving such values to $b, c, d$, and $e$, and also to $f$ and $g$, that the property which we have supposed may take place. 3. Let us represent these three couples of letters by the fractions $\frac{b}{c}$, $\frac{d}{e}$, and $\frac{f}{g}$; now, they ought to be such, that the difference of any two of them may be expressed by a fraction, whose numerator is 1. For since $\frac{b}{c} - \frac{d}{e} = \frac{be-de}{ce}$, this numerator, as we have seen, must be equal to $\pm 1$. Besides, one of these fractions is arbitrary; and it is easy to find another, in order that the given condition may take place. For example, let the first F F