Elements of Algebra

CHAP. VII. ADDITIONS. 571 81. *Example* 1. Let it be proposed to render rational the quantity, $\sqrt{30 + 62s - 7s^2}$, by taking only whole numbers for $s$. We shall here have to resolve this equation, $30 + 62s - 7s^2 = y^2$, which being multiplied by $7$, may be put into this form, $7 \times 30 + (31)^2 - (7s - 31)^2 = 7y^2$, or, making $7s - 31 = x$, and transposing, $x^2 = 1171 - 7y^2$, or $x^2 + 7y^2 = 1171$. This equation now comes under the case of Art. 64; so that we shall have $A = -7$, and $B = 1171$, from which we instantly perceive, that $y$ and $B$ must be prime to each other, since this last number contains no square factor. According to the method of Art. 65, we shall make $x = ny - 1171z$; and, in order that the equation may be resolvible, we must find for $n$ a positive, or negative integer, not $> B/2$; that is, not $> 580$, such that $n^2 - A$, or $n^2 + 7$, may be divisible by $B$, or by $1171$. I find $n = \pm 321$, which gives $n^2 + 7 = 1171 \times 88$; so that I substitute, in the preceding equation, $\pm 321y - 1171z$, instead of $x$; by which means, the whole is now divisible by $1171$, and when the division is performed, it becomes $88y^2 \mp 642yz + 1171z^2 = 1$. In order to resolve this equation, I shall employ the second method, explained in Art. 70, because it is in fact simpler and more convenient than the first. Now, as the coefficient of $y^2$ is less than that of $z^2$, we shall here have $D = 1171$, $D' = 88$, and $n = \pm 321$; wherefore retaining, for the sake of simplifying, the letter $y$, instead of $z$, and putting $y'$, instead of $z$, I shall make the following calculation, first supposing $n = 321$; $m = \frac{321}{88} = 4$, $n' = 321 - 4 \times 88 = -31$, $D'' = \frac{31^2+7}{88} = 11$, $y = 4y' + y''$, $m' = \frac{-31}{11} = -3$, $n'' = -31 + 3 \times 11 = 2$, $D''' = \frac{4+7}{11} = 1$, $y' = -3y'' + y'''$, $m'' = \frac{2}{1} = 2$, $n''' = 2 - 2 \times 1 = 0$, $D^{IV} = \frac{0+7}{1} = 7$, $y'' = 2y''' + y^{IV}$. .