Elements of Algebra

CHAP. VI. ADDITIONS. 547 In the Memoirs for 1770, and the following years, investigations will be found on the form of divisors of the numbers represented by $z^2 - Bq^2$; so that by the mere form of the number A, we shall often be able to judge of the impossibility of the equation $Ap^2 = z^2 - Bq^2$, where $Ay^2 + B = \square$, (Art. 52). *** CHAP. VI. Of Double and Triple Equalities. 61. We shall here say a few words on the subject of double and triple equalities, which are much used in the analysis of Diophantus, and for the solution of which, that great mathematician, and his commentators, have thought it necessary to give particular rules. When we have a formula, containing one or more unknown quantities, to make equal to a perfect power, such as a square, or a cube, &c. this is called, in the Diophantine analysis, a simple equality; and when we have two formulæ, containing the same unknown quantity, or quantities, to make equal each to a perfect power, this is called a double equality, and so on. Hitherto, we have seen how to resolve simple equalities, in which the unknown quantity does not exceed the second degree, and the power proposed is the square. Let us now see how double and triple equalities of the same kind are to be managed. 62. Let us first propose this double equality, $a + bx = \square$; $c + dx = \square$; where the unknown quantity is found only in the first degree. Making $a + bx = t^2$, and $c + dx = u^2$, and expunging $x$ from both equations, we have $ad - bc = dt^2 - bu^2$. Therefore, $dt^2 = bu^2 + ad - bc$; and, multiplying by $d$, $d^2t^2 = dbu^2 + (ad - bc)d$: so that the difficulty will be reduced to finding a rational value of $u$, such, that $dbu^2 + ad^2 - bcd$ may become a square. This simple NN2