Elements of Algebra

CHAP. XI. OF ALGEBRA. 393 We shall divide these numbers, as before, into simple and compound; the simple, or those which are not compounded of the preceding numbers, are these: $1, 2, 3, 11, 17, 19, 25, 41, 43, 49$, all which, except the squares $25$ and $49$, are prime numbers; and we may remark, in general, that, if a number is prime, and is not found in this series, we are sure to find its square in it. It may be observed, also, that all prime numbers contained in our formula, either belong to the expression $8n + 1$, or $8n + 3$; while all the other prime numbers, namely, those which are contained in the expressions $8n + 5$, and $8n + 7$, can never form the sum of a square and twice a square: it is farther certain, that all the prime numbers which are contained in one of the other formulæ, $8n + 1$, and $8n + 3$, are always resolvible into a square added to twice a square. 175. Let us proceed to the examination of the general formula $x^2 + cy^2$, and consider by what values of $x$ and $y$ we may transform it into a product of factors. We shall proceed as before; that is, we shall represent the formula by the product $(x + y \sqrt{-c}) \times (x - y \sqrt{-c})$, and shall likewise express each of these factors by two factors of the same kind; that is, we shall make $x + y \sqrt{-c} = (p + q \sqrt{-c}) \times (r + s \sqrt{-c})$, and $x - y \sqrt{-c} = (p - q \sqrt{-c}) \times (r - s \sqrt{-c})$; whence $x^2 + cy^2 = (p^2 + cq^2) \times (r^2 + cs^2)$. We see, therefore, that the factors are again of the same kind with the formula. With regard to the values of $x$ and $y$, we shall readily find $x = pr + cqs$, and $y = qr - ps$; or $x = pr - cqs$, and $y = ps + qr$; and it is easy to perceive how the formula may be resolved into a greater number of factors. 176. It will not now be difficult to obtain factors for the formula $x^2 - cy^2$; for, in the first place, we have only to write $-c$, instead of $+c$; but, farther, we may find them immediately in the following manner. As our formula is equal to the product $(x + y \sqrt{c}) \times (x - y \sqrt{c})$, let us make $x + y \sqrt{c} = (p + q \sqrt{c}) \times (r + s \sqrt{c})$, and $x - y \sqrt{c} = (p - q \sqrt{c}) \times (r - s \sqrt{c})$, and we shall immediately have $x^2 - cy^2 = (p^2 - cq^2) \times (r^2 - cs^2)$; so that this formula, as well as the preceding, is equal to a product whose factors resemble it in form. With regard to