History of Hindu Mathematics: A Source Book v2.c17.s14.80

v2.c17.s14.80

**Rational Solution.** In order to obtain a first solution of Nx²+1=y² the Hindus generally suggest the following tentative method: Take an arbitrary small rational number α, such that its square multiplied by the *guṇaka* N and increased or diminished by a suitably chosen rational number k will be an exact square. In other words, we shall have to obtain empirically a relation of the form $Na²± k=β²,$ where α, k, β are rational numbers. This relation will be hereafter referred to as the *Auxiliary Equation*. Then by Brahmagupta's Corollary, we get from it the relation $N(2αβ)²+k²=(β²+Nα²)²,$ or $N((2αβ)/k)²+1=((β²+Nα²)/k)².$