Elements of Algebra

CHAP. II. ADDITIONS. 517 the equation $Q^v = \mu^{iv} P^{iv} + Q^{iv}$, and then $P^v$ from this, $\frac{(Q^v)^2 - P^{iv} P^v}{P^{iv}} = \frac{1}{2}E$. Whence it is easy to draw this general conclusion, that, if $p^\lambda$ and $p^{\lambda+1}$ are the leading terms of the series $p', p'', p''', \ldots$ which are successively found with different signs, the term $p^{\lambda+1}$, and the following, will all return, after a certain number of intermediate terms, and it will be the same with the term $p^\lambda$, if we have $\frac{\pm p^\lambda}{p^{\lambda+1}}$ = or < 1. For let us imagine, as in Art. 34, that we have found $p^{\pi+2\xi} = p^\pi$, and $Q^{\pi+2\xi} = Q^\pi$, and suppose that $\pi$ is $>\lambda$, that is to say, $\pi = \lambda + \nu$; wherefore we may go back, on the one hand, from the term $p^\pi$ to the term $p^{\lambda+1}$, or $p^\lambda$, and on the other, from the term $p^{\pi+2\xi}$ to the term $p^{\lambda+2\xi+1}$, or $p^{\lambda+2\xi}$; and, as the terms from which we set out are equal on both sides, all the terms derived from them will likewise be respectively equal; so that we shall have $p^{\lambda+2\xi+1} = p^{\lambda+1}$, or even $p^{\lambda+\xi} = p^\lambda$, if $\frac{\pm p^\lambda}{p^{\lambda+1}}$ = or < 1. We may, therefore, judge beforehand of the beginning of the periods in the series $p^0, p', p'', p''', \ldots$ and consequently in the other series also, $q^0, q', q'', q''', \ldots$ $\mu, \mu', \mu'', \mu''', \ldots$ but as to the length of the periods, that depends on the nature of the number E, and entirely on the value of that number, as I could demonstrate, were I not afraid of being led into too long a detail. 37. Corollary 3. What we have demonstrated in the preceding corollary, may serve to prove the following theorem: Every equation of the form $p^2 - \kappa q^2 = 1$, (in which $\kappa$ is a positive integer number, but not a square, and $p$ and $q$ two indeterminate numbers) is resolvible in integer numbers. For, by comparing the formula $p^2 - \kappa q^2$ with the general formula, $Ap^2 + Bpq + cq^2$, we have $A=1$, $B=0$, $c=-\kappa$; wherefore $E = B^2 - 4AC = 4\kappa$, and $\frac{1}{2}\sqrt{E} = \sqrt{\kappa}$ (Art. 33). Wherefore, $p^0=1$, $q^0=0$; likewise $\mu < \sqrt{\kappa}$, $q'=\mu$, and $p' = \mu^2 - \kappa$; whence we see first, that $p'$ is negative, and consequently has a different sign from $p^0$; secondly, that $-p'$ is = or > 1, because $\kappa$ and $\mu$ are integer numbers; so that we shall have $\frac{p^0}{-p'}$ = or < 1; whence we shall find, from the preceding Article, $\lambda=0$, and $p^\xi = p^0 = 1$;