SOLUTION OF INDETERMINATE Hence b₁b-1 a whole number. a Now n²-N- (aik-b)-Na² a² aik²-2bkartk aª kaik-2ba1+1) a Therefore(ai²k-2ba+1) is a whole number. Since a. k have no common factor, it follows that ak-2ba+1_n²-N_ kan integer. a² Also k₁=¹² P W ai²k-2ba1+1 a² a₁²(b²-Na²)-2ba3+1 aib- (¹-¹)' - Nai. a Thus having known a single solution in positive integers of the equation Nxic-y², says, Brahmagupta, an infinite number of other integral solutions can be obtained by making use of the integral solutions of Nx²+1-y. If (p,q) be a solution of the former equation found empirically and if (a, 3) be an integral solution of the latter, then by the principle of Composition 259 x=plqa; p=q³±Npa will be a solution of the former. Repeating the operations, we can easily deduce as many solutions as we like. FORM Mnx+c=v²: In this connection, Brahmagupta says: If the remainder is that divided by a square, the first root is that divided by its root¹. This seems to mean that if we have the equation Mn²x+c=y² such that the multiplier (i.e. the 1. वर्गच्छिन्ने गुणके प्रथमं तन्मूल भाजितं भवति । coefficient of x) is divisible -BrSpSi. XVIII.70
