Elements of Algebra

552 ADDITIONS. CHAP. VII. We shall seek, therefore, the values of $n$, which may render $n^2 - \text{A}$ divisible by $\text{B}$; for this it will be sufficient, (Art. 47), to try for $n$ all the integer numbers, positive or negative, not $\ngtr \frac{\text{B}}{2}$; and if among these we find no one satisfactory, we shall at once conclude that it is impossible for $n^2 - \text{A}$ to be divisible by $\text{B}$, and therefore that the given equation is not resolvible in integer numbers. But if, in this manner, we find one, or more satisfactory numbers, we must take them, one after another, for $n$, which will give as many different equations, to be separately considered, each of which will furnish one, or more solutions, of the given question. With regard to such values of $n$ as would exceed that of $\frac{\text{B}}{2}$, we may neglect them, because they would give no equations different from those, which will result from the values of $n$ that are not $\ngtr \frac{\text{B}}{2}$, as we have already shewn (Art. 52). Lastly, as the condition from which we must determine $n$ is, that $n^2 - \text{A}$ may be divisible by $\text{B}$, it is evident, that each value of $n$ may be negative, as well as positive; so that it will be sufficient to try, successively, for $n$, all the natural numbers, that are not greater than $\frac{\text{B}}{2}$, and then to take the satisfactory values of $n$, both in plus and in minus. We have elsewhere given rules for facilitating the investigation of the values of $n$, that may have the property required, and even for finding those values à priori in a great number of cases. See the Memoirs of Berlin for the year 1767, pages 194, and 274. Resolution of the Equation $cy^2 - 2nyz + bz^2 = 1$, in Integer Numbers. This equation may be resolved by two different methods. First Method. 66. As the quantities $c, n, b$ are supposed to be integer numbers, as well as the indeterminate quantities $y$ and $z$, it is evident, that the quantity $cy^2 - 2nyz + bz^2$ must always be equal to integer numbers; consequently, unity will be its least possible value, unless it may become $0$, which can only happen, when this quantity may be resolved into two rational