Elements of Algebra

CHAP. VIII. ADDITIONS. 579 deduce others from it, by means of the equation $p^2 = Aq^2 + 1$; and that, frequently, at least when $B$ is not a prime number, there may be values of $x$ and $y$, which cannot be contained in the general expressions of M. Euler *. With regard to the manner of resolving equations of the form $p^2 = Aq^2 + 1$, I think that of Chap. VII., however ingenious it may be, is still far from being perfect. For, in the first place, it does not shew that every equation of this kind is always resolvible in whole numbers, when $a$ is a positive number not a square. Secondly, it is not demonstrated, that it must always lead to the solution sought for. Wallis, indeed, has professed to prove the former of these propositions; but his demonstration, if I may presume to say so, is a mere petitio principii. (See Chap. 99.) Mine, I believe, is the first rigid demonstration that has appeared; it is in the Melanges de Turin, Vol. IV.; but it is very long, and very indirect: that of Art. 37, is founded on the true principles of the subject, and leaves, I think, nothing to wish for. It enables us, also, to appreciate that of Chap. VII., and to perceive the inconveniences into which it might lead, if followed without precaution. This is what we shall now discuss. 86. From what we have demonstrated, Chap. II., it follows, that the values of $p$ and $q$, which satisfy the equation $p^2 - Aq^2 = 1$, can only be the terms of some one of the principal fractions derived from the continued fraction, which would express the value of $\sqrt{A}$; so that supposing this continued fraction to be represented thus, $\mu + \frac{1}{\mu' + \frac{1}{\mu'' + \frac{1}{\mu''' + ,~\&c.}}}$ we must have, $\frac{p}{q} = \mu + \frac{1}{\mu' + \frac{1}{\mu'' + ,~\&c.}}$ $+ \frac{1}{\mu^{\xi}};$ $\mu^{\xi}$ being any term whatever of the infinite series $\mu'$, $\mu''$, &c. the rank of which, $\xi$, can only be determined a posteriori. We must observe that, in this continued fraction, the numbers, $\mu$, $\mu'$, $\mu''$, &c. must all be positive, although we have * See Art. 45 of my Memoir on Indeterminate Problems, in the Memoirs of Berlin. 1767. PP2