Noble Number

A noble number is defined as an Irrational Number which has a Continued Fraction which becomes an infinite sequence of 1s at some point,

$$\nu\equiv [a_1, a_2, \ldots, a_n, \bar 1\,].$$

The prototype is the Golden Ratio $\phi$ whose Continued Fraction is composed entirely of 1s, $[\,\bar 1\,]$. Any noble number can written as

$$\nu ={A_n+\phi A_{n-1}\over B_n+\phi B_{n+1}},$$

where $A_k$ and $B_k$ are the Numerator and Denominator of the $k$th Convergent of $[a_1, a_2, \ldots, a_n]$. The noble numbers are a Subfield of $\Bbb{Q}(\sqrt{5}\,)$.

See also Near Noble Number

References

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, p. 236, 1979.

Schroeder, M. ``Noble and Near Noble Numbers.'' In Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 392-394, 1991.