Elements of Algebra

576 ADDITIONS. CHAP. VII. If we make $m=0$, we shall have $t=1$, and $u=0$; wherefore, $y=\pm 74$, or $=\pm 34$; and this last value is the least that will resolve the problem. I have already resolved this problem in the Memoirs of Berlin, for the year 1768, page 243; but as I have there employed a method somewhat different from the foregoing, and fundamentally the same as the *first* method of Art. 66, it was thought proper to repeat it here, in order that the comparison of the results, which are the same by both methods, might serve, if necessary, as a confirmation of them. 83. *Example 3.* Let it be proposed to find whole numbers, which being taken for $y$, may render rational the quantity, $\sqrt{(79y^2 + 101)}$. Here we shall have to resolve, in integers, the equation, $x^2 - 79y^2 = 101$, in which $y$ will be prime to 101, since this number does not contain any square factor. If we therefore suppose $x = ny - 101z$, $n^2 - 79$ must be divisible by 101, taking $n < \frac{101}{2} < 51$; we find $n = 33$, which gives $n^2 - 13 = 1010 = 101 \times 10$; thus, we may take $n = \pm 33$, and these will be the only values that have the condition required. Substituting, therefore, $\pm 33y - 101z$ instead of $x$, and then dividing the whole equation by 101, we shall have it transformed into $10y^2 \mp 66yz + 101z^2 = 1$. Let us, therefore, make $D' = 10$, $D = 101$, $n = \pm 33$, and first taking $n$ positive, we shall work as in the preceding example; thus, we shall have $m = \frac{33}{10} = 3$, $n' = 33 - 3 \times 10 = 3$, $D'' = \frac{9-79}{10} = -7$, $y = 3y' + y''$. Now, as $n' = 3$ is already $<\frac{D'}{2}$, and $<\frac{D''}{2}$, it is not necessary to proceed any farther: so that the equation will be transformed to this, $-7y'^2 - 6y'y'' + 10y''^2 = 1$, which being multiplied by $-7$, may be put into this form, $(7y' + 3y'')^2 - 79y''^2 = -7$. Since, therefore, $7$ is $<\sqrt{79}$, if this equation be resolvible, the number 7 must be found among the terms of the upper series of numbers answering to $\sqrt{79}$ in the Table (Art. 41), and also hold an even place there, since it has the sign $-$.