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Noble Number


A noble number nu is defined as an irrational number having a continued fraction that becomes an infinite sequence of 1s at some point,

 nu=[0,a_1,a_2,...,a_n,1^_].

The prototype is the inverse of the golden ratio phi^(-1), whose continued fraction is composed entirely of 1s (except for the a_0 term), [0,1^_].

Any noble number can be written as

 nu=(A_n+phi^(-1)A_(n-1))/(B_n+phi^(-1)B_(n+1)),

where A_k and B_k are the numerator and denominator of the kth convergent of [0,a_1,a_2,...,a_n].

The noble numbers are a subset of Q(sqrt(5)) but not a subfield, since there is no subfield lying properly between Q and Q(sqrt(5)). To see this, consider sqrt(5)=2phi-1, which must be contained in the same field as phi but is not a noble number since its continued fraction is [2,4^_].


See also

Near Noble Number, Periodic Continued Fraction

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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.

Referenced on Wolfram|Alpha

Noble Number

Cite this as:

Weisstein, Eric W. "Noble Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NobleNumber.html

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