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