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

230 BRAHMAGUPTA AS AN ALGEBRAIST The equations (2n+1) and (I. 2n+1) will then be absent and the equations (I. 2n-1) and (I. 2n) will be reduced respectively to xₙ₋₁ = q₂ₙ₋₁ y - c and xₙ = - c Giving an arbitrary integral value (t') to yₙ we get an in- tegral value of xₙ₋₁. Then proceeding backwards as before we calculate the values of x and y. Case (ii) : Next suppose that the mutual division is stopped after having obtained an even or odd number of quotients. Subcase (ii.1) : If the number of quotients obtained be even the reduced form of the original equation is r₂ y + 1 = r₂ₙ + 1 xₙ + c or yₙ ₊ ₁ = (r₂ₙ ₊ ₁ xₙ + c) / r₂ Giving a suitable integral value (t) to xₙ as will make yₙ₊₁ = (r₂ₙ₊₁ t + c) / r₂ₙ = an integral number, we get an integral value for yₙ by (2n+1). The values of x and y can then be calculated by proceeding as before. Subcase (ii.2) : If the number of quotients be odd the reduc- ed form of the quotient is r₂ₙ₋₁ xₙ = r₂ₙ yₙ - c or xₙ = (r₂ₙ yₙ - c) / r₂ₙ₋₁ Putting yₙ = t', where t' is an integer, such that xₙ = (r₂ₙ t' - c) / r₂ₙ₋₁ = a whole number, we get an integral value of xₙ ₋ ₁ by (2n). Whence can be calculated the values of x and y in integers. If x = α and y = β be the least integral solution of ax + c = by, we shall have aα + c = bβ Therefore a(bm + α) + c = b (am + β), m being any integer. Therefore, in general, x = bm + α But we have calculated before that