The principal thrust of this investigation is to provide families of quadratic polynomials fDkðXÞ 1⁄4 f 2 k X 2 þ 2ekX þ CgkAN; where ek f 2 k C 1⁄4 n (for any given nonzero integer n) satisfying the property that for any XAN; the period length ck 1⁄4 cð ffiffiffiffiffiffiffiffiffiffiffiffiffiffi DkðXÞ p Þ of the simple continued fraction expansion of… (More)

The purpose of this article is to provide criteria for the solvability of the Diophantine equation a2X2− bY 2 = c in terms of the simple continued fraction expansion of √ a2b, and to explore criteria for the solvability of AX2 − BY 2 = C for given A, B, C ∈ N in the general case. This continues work in [9]–[11].

Mathematics is the queen of sciences and number theory is the queen of mathematics. But why is it computer science? It turns out to be critical for cryptography!