Elements of Algebra

CHAP. VII. ADDITIONS. 553 factors. As this case is attended with no difficulty, we shall at once neglect it, and the question will be reduced to finding such values of $y$ and $z$, as will make the quantity in question the least possible. If the minimum be equal to unity, we shall have the resolution of the proposed equation; otherwise, we shall be assured, that it admits of no solution in integer numbers. So that the present problem falls under the third problem of Chap. II., and admits of a similar solution. Now, as we have here $(2n)^2 - 4BC = 4A$ (Art. 65), we must make two distinct cases, according as A shall be positive, or negative. First case, when $n^2 - BC = A < 0$. 67. According to the method of Art. 32, we must reduce the fraction $\frac{n}{c}$, taken positively, to a continued fraction; this may be done by the rule of Art. 4; then, by the formulae of Art. 10, we shall form the series of fractions converging towards $\frac{n}{c}$, and shall have only to try, successively, the numerators of those fractions for the number $y$, and the corresponding denominators for the number $z$: if the given formula be resolvable in integers, we shall in this way find the satisfactory values of $y$ and $z$; and, conversely, we may be certain, that it admits not of any solution in integer numbers, if no satisfactory values are found among the numbers that are tried. Second case, when $n^2 - BC = A > 0$. 68. We shall here employ the method of Art. 33 et seq. so that, because $E = 4A$, we shall at once consider the quantity (Art. 39), $a = \frac{n \pm \sqrt{A}}{c}$, in which we must determine the signs both of the value of $n$, which we have seen may be either positive or negative, and of $\sqrt{A}$, so that it may become positive; we shall then make the following calculation: $Q^0 = -n$, $P^0 = c$, $\mu < \frac{-Q^0 \pm \sqrt{A}}{P^0}$ $Q' = -\mu P^0 + Q^0$, $P' = \frac{Q'^2 - A}{P^0}$, $\mu' < \frac{-Q' \mp \sqrt{A}}{P'}$