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

CAKRAVĀLA OR CYCLIC METHOD 257
= {½(β² + 2)}² (ii)
Again applying the Corollary, we get
N {½αβ(β² + 2)}² + 1 = {½(β⁴ + 4β² + 2)}² (iii)
Now by the Lemma we obtain from (ii) and (iii)
N {½αβ(β² + 3) (β² + 1)}² + 1
= [(β² + 2){½(β² + 3) (β² + 1) − 1}]²
Hence x = ½αβ(β² + 3) (β² + 1),
y = (β² + 2){½(β² + 3) (β² + 1) − 1}
is a solution of Nx² + 1 = y²
This can be proved without difficulty that these values of x
and y are integral. Since if β is even, β² + 2 is also even. And
hence the above values of x and y are integral. On the other
hand, if β is odd, β² is also odd; under these conditions β² + 1 and
β² + 3 are even. In this also, therefore, the above values must be
integral.
Putting p = αβ, q = β² + 2, we can write the above solution
in the form
x = ½p (q² − 1).
y = ½q (q² − 3).
This was the form in which the solution was found by
Euler.
Cakravāla or Cyclic Method
We have shown in the preceding articles that the most
fundamental step in Brahmagupta's method for the general
solution in positive integers of the equation
Nx² + 1 = y²
where N is a non-square integer, is to form an auxiliary equation
of the kind
Na² + k = b²
where a and b are positive integers and k = ±1, ±2 or ±4. From
this auxiliary equation, by the Principle of Composition, applied
repeatedly whenever necessary, one can derive, as we have alrea-
dy shown above, one positive integral solution of the original
Varga-prakṛti or Square-Nature. And thence again, by means
of the same principle, an infinite number of other solutions in
integers can be obtained. How to form an auxiliary equation of