Elements of Algebra

458 ELEMENTS PART II. become equal, which does not enter into the subject of the problem. 248. Question 3. Required in general three cubes, $x^3$, $y^3$, and $z^3$, whose sum may be equal to a cube. We have seen that two of these cubes may be supposed to be known, and that from them we may determine the third, provided the two are not equal; but the preceding method furnishes in each case only one value for the third cube, and it would be difficult to deduce from it any new ones. We shall now, therefore, consider the three cubes as unknown; and, in order to give a general solution, let us make $x^3 + y^3 + z^3 = v^3$. Here, by transposing one of the terms, we have $x^3 + y^3 = v^3 - z^3$, the conditions of which equation we may satisfy in the following manner. 1. Let $x = p + q$, and $y = p - q$, and we shall have, as before, $x^3 + y^3 = 2p(p^2 + 3q^2)$. Also, let $v = r + s$, and $z = r - s$, which gives $v^3 - z^3 = 2s(s^2 + 3r^2)$; therefore we must have $2p(p^2 + 3q^2) = 2s(s^2 + 3r^2)$, or $p(p^2 + 3q^2) = s(s^2 + 3r^2)$. 2. We have already seen (Art. 176), that a number, such as $p^2 + 3q^2$, can have no divisors except numbers of the same form. Since, therefore, these two formulæ, $p^2 + 3q^2$, and $s^2 + 3r^2$, must necessarily have a common divisor, let that divisor be $t^2 + 3u^2$. 3. And let us, therefore, make $p^2 + 3q^2 = (f^2 + 3g^2) \times (t^2 + 3u^2)$, and $s^2 + 3r^2 = (h^2 + 3k^2) \times (t^2 + 3u^2)$, and we shall have $p = ft + 3gu$, and $q = gt - fu$; consequently, $p^2 = f^2t^2 + 6fgtu + 9g^2u^2$, and $q^2 = g^2t^2 - 2fgtu + f^2u^2$; whence, $p^2 + 3q^2 = (f^2 + 3g^2)t^2 + (3f^2 + 9g^2)u^2$; or $p^2 + 3q^2 = (f^2 + 3g^2) \times (t^2 + 3u^2)$. 4. In the same manner, we may deduce from the other formula, $s = ht + 3ku$, and $r = kt - hu$; whence results the equation, $(ft + 3gu) \times (f^2 + 3g^2) \times (t^2 + 3u^2) = (ht + 3ku) \times (h^2 + 3k^2) \times (t^2 + 3u^2)$, which being divided by $t^2 + 3u^2$, and reduced, gives $ft(f^2 + 3g^2) + 3gu(f^2 + 3g^2) = ht(h^2 + 3k^2) + 3ku(h^2 + 3k^2)$, or $ft(f^2 + 3g^2) - ht(h^2 + 3k^2) = 3ku(h^2 + 3k^2) - 3gu(f^2 + 3g^2)$, by which means $t = \frac{3k(h^2+3k^2)-3g(f^2+3g^2)}{f(f^2+3g^2)-h(h^2+3k^2)}u$.