Elements of Algebra

CHAP. VII. ADDITIONS. 561 $y = pt - (Bq - np)u,$ $z = qt + (cp - nq)u.$ 73. The whole therefore is reduced to resolving the equation $t^2 - Au^2 = 1$. Now, 1st, if $A$ be a negative number, it is evident, that this equation cannot subsist, in whole numbers, except by making $u = 0$, and $t = 1$, which would give $y = p$, and $z = q$. Whence we may conclude that, in the case of $A$ being a negative number, the proposed equation, $cy^2 - 2nyz + Bz^2 = 1$, can never admit but of one solution in whole numbers. The case would be the same, if $A$ were a positive square number; for making $A = a^2$, we should have $(t + au) \times (t - au) = 1$; wherefore, $t + au = \pm 1$, and $t - au = \pm 1$; wherefore, $2au = 0$, $u = 0$, and consequently $t = \pm 1$. 2dly. But if $A$ be a positive number, not square, then the equation, $t^2 - Au^2 = 1$, is always capable of an infinite number of solutions, in whole numbers, (Art. 37), which may be found by the formulæ already given (Art. 71); but it will be sufficient to find the least values of $t$ and $u$; and, for this purpose, as soon as we have arrived, in the series $p', p'', p'''$, &c. at a term equal to unity, we shall have only to calculate, by the formulæ of Art. 25, the corresponding terms of the two series $p', p'', p'''$, &c. and $q', q'', q'''$, &c. for these will be the values required of $t$ and $u$. Whence it is evident, that the same calculation made for resolving the equation $v^2 - A\xi^2 = M$, will serve also for the equation $t^2 - Au^2 = 1$. Provided that $A$ does not exceed 100, we have the least values of $t$ and $u$ calculated in the Table, at the end of Chap. VII. of the preceding Treatise, and in which the numbers $a, m, n$, are the same as those that are here called $A, t$ and $u$. 74. Let us denote by $t', u'$, the least values of $t, u$, in the equation $t^2 - Au^2 = 1$; and in the same manner as these values may serve to find new values of $y$ and $z$, in the equation, $cy^2 - 2nyz + Bz^2 = 1$, so they will likewise serve for finding new values of $t$ and $u$ in the equation $t^2 - Au^2 = 1$, which is only a particular case of the former. For this purpose, we shall only have to suppose $c = 1$, and $n = 0$, which gives $-B = A$, and then take $t, u$, instead of $y, z$, and $t', u'$, instead of $p, q$. Making these substitutions, therefore, in the general expressions of $y$ and $z$ (Art. 72), and farther, putting $\tau, v$, instead of $t, u$, we shall have, generally, oo