History of Hindu Mathematics: A Source Book v2.c17.s14.85

v2.c17.s14.85

If m² be a rational number optionally chosen, we have the identity $N.1²+(m²-N)=m²,$ or $N.1²-(N-m²)=m².$ Then, applying Brahmagupta's Corollary to either, we get $N(2m)²+(m²~ N)²=(m²+N)²;$ $∴ N(2m/(m²~ N))²+1=((m²+N)/(m²~ N))².$ Hence $x = 2m/(m²~ N), y = (m²+N)/(m²~ N),(B)$ where m is any rational number, is a solution of the equation $Nx²+1=y².$