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

254 BRAHMAGUPTA AS AN ALGEBRAIST Let us solve the first example 8x²＋1＝y². We assume the optional number to be 3. Its square is 9; the prakṛti of multiplier is 8, their difference is 9–8＝1. Dividing by this twice the optional number (2×3, i.e. 6), namely 6, we get the lesser root for the addi- tive unity as 6. Whence proceeding as before, we get the greater to be 17. Thus here x＝6 and y＝17. Let us use this method for the equation 11x²＋1＝y². Let the optional number be 3. Its square is 9: multiplier or prakṛti is 11; the difference is 11–9＝2; dividing by this twice the optional number (2×3), namely 6, we get 6/2＝3, which is the lesser root. Consequently the greater root would be 10. Thus for this equation x＝3 and y＝10. Solution in Positive Integers The Indian algebraists usually aimed at obtaining solutions of the varga-prakṛti or Square-nature in positive integers or abhinna. The tentative methods of Brahmagupta and Śrīpati always did not furnish solutions in positive integers. These auth- ors, however, discovered that if the interpolator of auxiliary equa- tion in the tentative method be ±1, ±2 or ±4, an integral solu- tion of the equation Nx²＋1＝y² can always be found. Thus Śrīpati says : If 1, 2 or 4 be the additive or subtractive (of the auxi- liary equation), the lesser and greater roots will be integral (abhinna)¹. (i) If k＝±1, then the auxiliary equation will be Nα²±1＝β where α and β are intergers. Then by Brahmagupta' Corollary we get x＝2αβ and y＝β²＋Nα² as the required first solution in positive integers of the equation Nx²＋1＝y²

1. इष्टवर्गं प्रकृत्योर्यद्विवरं तेन वा भजेत् । द्विघ्नमिष्टं कनिष्ठं तत् पदं स्यादेक संयुतौ । ततो ज्येष्ठमिहानन्त्यं भावनाभिस्तथेष्टतः ।। Bījagaṇita, Varga-Prakṛti, 5-6