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