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