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

250 BRAHMAGUPTA AS AN ALGEBRAIST
Brahmagupta's Corollary also follows at once from the
above by putting α'=α, β'=β and k'=k.
N (2αβ)²+k²=(β²±Nα²)²
Thus the roots are x=2αβ and y=β²±Nα² which is the
Corollary.
It would be seen that modern historians of mathematics are
incorrect when they say that Fermat (1657) was the first to
state that the equation Nx²+1=y², where N is a non-square
integer has an unlimited number of solutions in integers. For
this assertion, history takes us to the early Seventh Century A.D.
when Brahmagupta wrote his classical treatise, the Brāhmasphu-
tasiddhānta, and gave the well known two Lemmas and the
Corollary to the first Lemma.
Second Lemma of Brahmagupta
In the Brāhmasphuṭa siddhānta, we find another important
Lemma by Brahmagupta stated as follows :
On dividing the two roots (of a square- Nature) by the
square-root of its additive or subtractive, the roots for
interpolator unity (will be found).¹
This Lemma when expressed in the modern language of
algebra would mean that if x=α, y=β be a solution of the
equation.
Nx²+k²=y²
then x=α/k, y=β/k is a solution of the equation
Nx²+1=y².
This rule, at another place, has been re-enunciated as
follows :
If the interpolator is that divided by a square then
the roots will be those multiplied by its square-
root.²

1. प्रक्षेपशोधक हृते मूले प्रक्षेपके रूपे ।
   —BrSpSi. XVIII. 65
2. वर्गाच्छिन्ने क्षेपे तत्पदगुणिते तदा मूले ।
   —BrSpSi. XVIII. 70