Elements of Algebra

CHAP. XIII. OF ALGEBRA. 403 shew, that $x^4 - y^4$ can no longer become a square, when one of the biquadrates is even, and the other odd. 1. We may observe, if the first term, $x^4$, were even, and $y^4$ odd, the impossibility of the thing would be self-evident, since we should have a number of the form $4n+3$; which cannot be a square: therefore, let $x$ be odd, and $y$ even; then $x^2 = p^2 + q^2$, and $y^2 = 2pq$; whence $x^4 - y^4 = p^4 - 2p^2q^2 + q^4 = (p^2 - q^2)^2$, where one of the two numbers $p$ and $q$ must be even, and the other odd. 2. Now, as $p^2 + q^2 = x^2$ must be a square, we have $p = r^2 - s^2$, and $q = 2rs$; whence $x = r^2 + s^2$: but from that results $y^2 = 2(r^2 - s^2) \times 2rs$, or $y^2 = 4rs \times (r^2 - s^2)$, and as this must be a square, its fourth part, $rs(r^2 - s^2)$, whose factors are prime to each other, must likewise be a square. 3. Let us, therefore, make $r = t^2$, and $s = u^2$, and we shall have the third factor $r^2 - s^2 = t^4 - u^4$, which must also be a square. Now, as this factor is equal to the difference of two biquadrates, which are much less than the first, the preceding demonstration is fully confirmed; and it is evident, that, if the difference of two biquadrates could become equal to the square of a number (however great we may suppose it), we could, by means of this known case, arrive at differences less and less, which would also be reducible to squares, without our being led back to the two evident cases mentioned at first. It is impossible, therefore, for the thing to take place even with respect to the greatest numbers. 208. The first part of the preceding demonstration, namely, where $x$ and $y$ are supposed odd, may be abridged as follows: if $x^4 - y^4$ were a square, we must have $x^2 = p^2 + q^2$, and $y^2 = p^2 - q^2$, representing by $p$ and $q$ numbers, the one of which is even and the other odd; and by these means we should obtain $x^2y^2 = p^4 - q^4$; and, consequently, $p^4 - q^4$ must be a square. Now, this is a difference of two biquadrates, the one of which is even and the other odd; and it has been proved, in the second part of the demonstration, that such a difference cannot become a square. 209. We have therefore proved these two principal propositions; that neither the sum, nor the difference, of two biquadrates, can become a square number, except in a very few self-evident cases. Whatever formulæ, therefore, we wish to transform into squares, if those formulæ require us to reduce the sum, or the difference of two biquadrates to a square, it may be pronounced that the given formulæ are likewise impossible;