6. Pell’s Equation - Deepstash

6. Pell’s Equation

Brahmagupta studied this equation 1000 years before Pell’s birth. Pell’s equation is of form, nx² + 1 = y², which can also be written as y² – nx² = 1, where ‘n’ is an integer and we have to solve it for (x, y) integer solutions. Brahmagupta also provided a lemma, in which he stated that if (a, b) and (c, d)

are integer solutions of ‘Pell type equations’ of the form na²+ k = b² and nc² + k’ = d²

then, (bc + ad, bd + nac) and (bc – ad, bd – nac)are both integer solutions of the ‘Pell type equation’.

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