Elements of Algebra

540 ADDITIONS. CHAP. V. be the quotient of this division, so that $AA' = n^2 - B$, and the equation, $Ap^2 = z^2 - Bq^2 = (n^2 - B)q^2 - 2nAqq' + A^2q'^2$, being divided by $A$, will become $p^2 = A'q^2 - 2nqq' + Aq'^2$, where $A'$ will necessarily be less than $A$, because $A' = \frac{n^2 - B}{A}$, and $B < A$, and $n$ not $> \frac{A}{2}$. First, if $A'$ be a square number, it is evident this equation will be resolvible by the known methods; and the simplest solution will be obtained, by making $q' = 0$, $q = 1$, and $p = \sqrt{A'}$. Secondly, if $A'$ be not a square, we must ascertain whether it be less than $B$, or at least whether it be divisible by any square number, so that the quotient may be less than $B$, abstracting from the signs; then we must multiply the whole equation by $A'$, and, because $AA' = n^2 - B$, we shall have $A'p^2 = (A'q - nq')^2 - Bq'^2$; so that $Bq'^2 + A'p^2$ must be a square; hence, dividing by $p^2$, and making $\frac{q'}{p} = y$, and $A' = c$, we shall have to make a square of the formula $By^2 + c$, which evidently resembles that of Art. 52. Thus, if $c$ contains a square factor $\gamma^2$, we may suppress it, by multiplying the value which we shall find for $y$ by $\gamma$, in order to have its true value; and we shall have a formula similar to that of Art. 51, but with this difference, that the coefficients, $B$ and $c$, of our last will be less than the coefficients, $A$ and $B$, of the other. 54. But if $A'$ be not less than $B$, nor becomes so when divided by the greatest square, which measures it, then we must make $q = vq' + q''$; and, substituting this value in the equation, it will become $p^2 = A'q''^2 - 2n'q''q' + A''q'^2$, where $n' = n - vA'$, and $A'' = A'v^2 - 2nv + A = \frac{n'^2 - B}{A'}$. We must determine the whole number $v$, which is always possible, so, that $n'$ may not be $> \frac{A'}{2}$, abstracting from the