Elements of Algebra

CHAP. XI. OF ALGEBRA. 389 into two rational factors; and here particular artifices are necessary, in order to find such values for $x$ and $y$, that the formula may contain two, or more factors. We shall, however, render this inquiry less difficult by observing, that our formula may be easily transformed into another, in which the middle term is wanting; for we have only to suppose $x = \frac{z-by}{2a}$, in order to have the following formula; $\frac{z^2-2byz+b^2y^2}{4a} + \frac{byz-b^2y^2}{2a} + cy^2 = \frac{z^2+(4ac-b^2)y^2}{4a}$; so that, neglecting the middle term, we shall consider the formula $ax^2 + cy^2$, and shall seek what values we must give to $x$ and $y$, in order that this formula may be resolved into factors. Here it will be easily perceived, that this depends on the nature of the numbers $a$ and $c$; so that we shall begin with some determinate formulæ of this kind. 168. Let us, therefore, first propose the formula $x^2 + y^2$, which comprehends all the numbers that are the sum of two squares, the least of which we shall set down; namely, those between 1 and 50: 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50. Among these numbers there are evidently some prime numbers which have no divisors, namely, the following: 2, 5, 13, 17, 29, 37, 41: but the rest have divisors, and illustrate this question, namely, ‘What values are we to adopt for $x$ and $y$, in order that the formula $x^2 + y^2$ may have divisors, or factors, and that it may have any number of factors?’ We shall observe, farther, that we may neglect the cases in which $x$ and $y$ have a common divisor, because then $x^2 + y^2$ would be divisible by the same divisor, and even by its square. For example, if $x = 7p$ and $y = 7q$, the sum of the squares, or $49p^2 + 49q^2 = 49(p^2 + q)$, will be divisible not only by 7, but also by 49: for which reason, we shall extend the question no farther than the formulæ, in which $x$ and $y$ are prime to each other. We now easily see where the difficulty lies: for though it is evident, when the two numbers $x$ and $y$ are odd, that the formula $x^2 + y^2$ becomes an even number, and, consequently, divisible by 2; yet it is often difficult to discover whether the formula have divisors or not, when one of the numbers is even and the other odd, because the formula itself, in that case, is also odd. We do not mention the case in which $x$ and $y$