ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

256 BRAHMAGUPTA AS AN ALGEBRAIST Substituting the value of N in the right-hand side expres- sion from (i), we have N.( αβ / 2 )² + 1 = ( (β²-2) / 2 )² (iii) Composing (ii) and (iii), N { α/2 (β²-1) }² + 1 = { β/2 (β²-3) }² Hence x= ½ αβ, y= ½ (β²--2); and x= ½α(β²-1), y= ½β(β²-3); are solutions of Nx²+1=y². If β be even, the first values of (x,y) are integral. If β be odd, the second values are integral. (iv) Finally, suppose k=-4; the auxiliary equation is Nα²-4 = β² Then the required first solution in positive integers of Nx²+1=y² is x= ½αβ(β²+3) (β²+1) y=(β²+2) { ½(β²+3) (β²+1)-1 }. Brahmagupta says : In the case of 4 as subtractive, the square of the second is increased by three and by unity; half the product of these sums and that as diminished by unity (are obtained). The latter multiplied by the first sum less unity is the (required) second root; the former multi- plied by the product of the (old) roots will be the first root corresponding to the (new) second root.¹ The rationale of this solution, as given by Datta and Singh is as follows : Nα²-4=β² (i) N(α/2)²-1=(β/2)² Hence by Brahmagupta's Corollary, we get N ( αβ / 2 )² + 1 = ( β²/4 + N α²/4 )²

1. चतुरूनेऽन्यपद कृती त्र्येकयुते वधदलं पृथग्व्येकम् । व्येकाद्वाहतमन्त्यं पदवध गुणमाद्यमान्त्यपदम् ॥ BrSpSi. XVIII. 68