Elements of Algebra

CHAP. XI. OF ALGEBRA. 395 returns to the formula $x^2 + cy^2$, and which we have already considered, if multiplied together, will produce a number of the same form. It only remains, therefore, to examine to what formula we are to refer the product of two numbers of the first kind, or of the form $ax^2 + cy^2$. For this purpose, let us multiply the two formulæ $(ap^2 + cq^2) \times (ar^2 + cs^2)$, which are of the first kind. It is easy to see that this product may be represented in the following manner: $(apr + cqs)^2 + ac(ps - qr)^2$. If, therefore, we suppose $apr + cqs = x$, and $ps - qr = y$, we shall have the formula $x^2 + acy^2$, which is of the last kind. Whence it follows, that if two numbers of the first kind, $ax^2 + cy^2$, be multiplied together, the product will be a number of the second kind. If we represent the numbers of the first kind by I, and those of the second by II, we may represent the conclusion to which we have been led, abridged as follows: I $\times$ I gives II; I $\times$ II gives I; II $\times$ II gives II. And this shews much better what the result ought to be, if we multiply together more than two of these numbers; namely, that I $\times$ I $\times$ I gives I; that I $\times$ I $\times$ II gives II; that I $\times$ II $\times$ II gives I; and lastly, that II $\times$ II $\times$ II gives II. 180. In order to illustrate the preceding Article, let $a = 2$, and $c = 3$; there will result two kinds of numbers, one contained in the formula $2x^2 + 3y^2$, the other contained in the formula $x^2 + 6y^2$. Now, the numbers of the first kind, as far as 50, are 1st, 2, 3, 5, 8, 11, 12, 14, 18, 20, 21, 27, 29, 30, 32, 35, 44, 45, 48, 50; and the numbers of the second kind, as far as 50, are 2d, 1, 4, 6, 7, 9, 10, 15, 16, 22, 24, 25, 28, 31, 33, 36, 40, 42, 49. If, therefore, we multiply a number of the first kind, for example, 35, by a number of the second, suppose 31, the product 1085 will undoubtedly be contained in the formula $2x^2 + 3y^2$; that is, we may find such a number for $y$, that $1085 - 3y^2$ may be the double of a square, or $= 2x^2$: now, this happens, first, when $y = 3$, in which case $x = 23$; in the second place, when $y = 11$, so that $x = 19$; in the third place, when $y = 13$, which gives $x = 17$; and, in the fourth place, when $y = 19$, whence $x = 1$. We may divide these two kinds of numbers, like the others, into simple and compound numbers: we shall apply