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## Main Question or Discussion Point

Let [tex]Q(\sqrt{k})[/tex], for some positive integer k, be the extension of the field of rationals with basis [tex](1, \sqrt{k})[/tex]. For example, in [tex]Q(\sqrt{5})[/tex] the element [tex]({1 \over 2}, {1 \over 2})[/tex] is the golden ratio = [tex]{1 \over 2} + {1 \over 2}\sqrt{5}[/tex].

Given an extension [tex]Q(\sqrt{k})[/tex], let [tex]B(n)[/tex] denote the 'Binet formula',

Given an extension [tex]Q(\sqrt{k})[/tex], let [tex]B(n)[/tex] denote the 'Binet formula',

[tex]B(n) = {{p^n - (1-p)^n} \over \sqrt k}[/tex],

where [tex]p = ({1 \over 2}, {1 \over 2})[/tex].*n*= 0, 1, 2, ...__Conj. 1__: [tex]B(n)[/tex] produces only integers, iif [tex]k \equiv 1 \ ("mod" \ 4)[/tex].__Conj. 2__: When [tex]k \equiv 1 \ ("mod" \ 4)[/tex], [tex]B(n)[/tex] is the closed-form formula for the recurrence sequence[tex]x_n = x_{n-1} + A \; x_{n-2}[/tex],

*n*= 2, 3, 4, ...[tex]x_0 = 0, \ \ x_1 = 1[/tex]

with [tex]A = {{k - 1} \over 4}[/tex].