Elements of Algebra

CHAP. XI. OF ALGEBRA. 391 factors in all, and the number might be augmented by this method to any length. 171. As in this solution we have found only the second powers of $p$, $q$, $r$, and $s$, we may also take these letters minus. If $q$, for example, be negative, we shall have $x = pr + qs$, and $y = ps - qr$; but the sum of the squares will be the same as before; which shews, that when a number is equal to a product, such as $(p^2 + q^2) \times (r^2 + s^2)$, we may resolve it into two squares in two ways; for we have first found $x = pr - qs$, and $y = ps + qr$, and then also $x = pr + qs$, and $y = ps - qr$. For example, let $p = 3$, $q = 2$, $r = 2$, and $s = 1$: then we shall have the product $13 \times 5 = 65 = x^2 + y^2$; in which $x = 4$, and $y = 7$; or $x = 8$, and $y = 1$; since in both cases $x^2 + y^2 = 65$. If we multiply several numbers of this class, we shall also have a product, which may be the sum of two squares in a greater number of ways. For example, if we multiply together $2^2 + 1^2 = 5$, $3^2 + 2^2 = 13$, and $4^2 + 1^2 = 17$, we shall find 1105, which may be resolved into two squares in four ways, as follows: 1. $33^2 + 4^2$, 2. $32^2 + 9^2$, 3. $31^2 + 12^2$, 4. $24^2 + 23^2$. 172. So that among the numbers that are contained in the formula $a^2 + y^2$, are found, in the first place, those which are, by multiplication, the product of two or more numbers, prime to each other; and, secondly, those of a different class. We shall call the latter *simple factors* of the formula $a^2 + y^2$, and the former *compound factors*; then the simple factors will be such numbers as the following: 1, 2, 5, 9, 13, 17, 29, 37, 41, 49, &c. and in this series we shall distinguish two kinds of numbers; one are prime numbers, as 2, 5, 13, 17, 29, 37, 41, which have no divisor, and are all (except the number 2), such, that if we subtract 1 from them, the remainder will be divisible by 4; so that all these numbers are contained in the expression $4n + 1$. The second kind comprehends the square numbers 9, 49, &c. and it may be observed, that the roots of these squares, namely, 3, 7, &c. are not found in the series, and that their roots are contained in the formulæ $4n - 1$. It is also evident, that no number of the form $4n - 1$ can be the sum of two squares; for since all numbers of this form are odd, one of the two squares must be even, and the other odd. Now, we have already seen, that all even squares are divisible by 4, and that the odd squares are contained in the formula $4n + 1$: if we therefore add