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