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

BHĀSKARA I AND KUṬṬAKA OPERATIONS 233 furnished by the above and applying the method of such quadratics, obtain the (simplest) solution by the substitution of 2, 3 etc. successively in the general solu- tion). Then calculate the ahargaṇa and the revolu- tions performed by Saturn and Mars in that time toge- ther with the number of solar years elapsed.¹ Let x and y denote the residues of the revolution of Mars and Saturn respectively. Then we have to find out two numbers x and y such that each of the expressions x+y, x-y and xy+1 may be a perfect square. Let x+y=4P² and x-y=4Q², so that x=2P²+2Q² y=2P²-2Q² and therefore xy+1=(2P²-1)²+4(P²-Q⁴) Hence the condition that xy+1 be a perfect square is that P²=Q⁴. Substituting these values, we have x=2(Q⁴+Q²) y=2(Q⁴-Q²) where Q may possess any of the values 2, 3, 4,......but not 1. (We neglect the case when x or y is zero).

1. भाज्यं न्यसेदुपरि हारमधश्च तस्य । खण्ड्यात्परस्परमथो विनिधाय लब्धम् । केनाऽऽहतोऽ