History of Hindu Mathematics: A Source Book v2.c17.s15.39

v2.c17.s15.39

¹ *Hankel's Proof*: Hankel proves these two results thus : Since a₁k = an + b and k = b² - Na², we get a₁(b² - Na²) = an + b, or b/a(a₁b - 1) = (n + Naa₁). Since a, b have no common factor, a must divide a₁b - 1; that is, $(a₁b - 1)/a = an integer.$