Elements of Algebra

558 ADDITIONS. CHAP. VII. not resolvible in integers, unless M be found equal to unity, and positive. If this condition take place, then we make $\xi = 0$, $\psi = \pm 1$, and go back from these values to those of $y$ and $z$. This method is founded on the same principles as that of Art. 67; but it has the advantage of not requiring any trial. 2dly. Let A be now a positive number, and we shall have $A = N^2 - LM$. And as $N^2$ cannot be greater than $\frac{LM}{4}$, it is evident that the equation cannot subsist, unless $-LM$ be a positive number; that is to say, unless L and M have contrary signs. Thus, A will necessarily be $< -LM$, or at farthest $=-LM$, if $N=0$; so that we shall have $-LM =$, or $< A$; and, consequently, $M^2 =$, or $< A$; or $M =$, or $< \sqrt{A}$. The case of $M = \sqrt{A}$ cannot take place, except when A is a square; consequently, this case may be easily resolved by the method already given (Art. 69). There remains, now, only the case in which A is not a square, and in which we shall necessarily have $M < \sqrt{A}$ (abstracting from the sign of M); then the equation, $u^2 - A\xi^2 = M$, will come under the case of the theorem, Art. 38, and may therefore be resolved by the method there explained. Hence, we have only to make the following calculation: $Q^0 = 0$, $P^0 = 1$, $\mu < \sqrt{A}$ $Q' = \mu$, $P' = Q'^2 - A$, $\mu' < \frac{-Q' - \sqrt{A}}{P'}$ $Q'' = \mu'P' + Q'$, $P'' = \frac{Q''^2 - A}{P'}$, $\mu'' < \frac{-Q'' + \sqrt{A}}{P''}$ $Q''' = \mu''P'' + Q''$, $P''' = \frac{Q'''^2 - A}{P''}$, $\mu''' < \frac{-Q''' - \sqrt{A}}{P'''}$ $\ldots$ $\ldots$ $\ldots$ continuing it until two corresponding terms of the first and second series appear again together; or until in the series $P', P'', P''', \ldots$ there be found a term equal to unity, and positive; that is to say, $= P^0$: for then all the succeeding terms will return in the same order in each of the three series (Art. 37). If in the series $P', P'', P''', \ldots$ there be found a term equal to M, we shall have the resolution of the given