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Date: 30-1-2021
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Date: 2-3-2020
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A noble number is defined as an irrational number having a continued fraction that becomes an infinite sequence of 1s at some point,
The prototype is the inverse of the golden ratio , whose continued fraction is composed entirely of 1s (except for the term), .
Any noble number can be written as
where and are the numerator and denominator of the th convergent of .
The noble numbers are a subset of but not a subfield, since there is no subfield lying properly between and . To see this, consider , which must be contained in the same field as but is not a noble number since its continued fraction is .
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.
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