History of Hindu Mathematics: A Source Book v2.c17.s14.116

v2.c17.s14.116

**Solution in Positive Integers.** As has been stated before, the aim of the Hindus was to obtain solutions of the Square-nature in positive integers; so its first solution must be integral. But neither the tentative method of Brahmagupta nor that of Śrîpati is of much help in this direction, for they do not *always* yield the desired result. These authors, however, discovered that if the interpolator of the auxiliary equation in the tentative method be ± 1, ± 2 or ± 4, an integral solution of the equation Nx²+1=y² can always be found. Thus Śrîpati (1039) expressly observes, "If 1, 2 or 4 be the additive or subtractive (of the auxiliary equation) the lesser and greater roots will be integral (*abhinna*).²