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Near Noble Number


A near noble number is a real number 0<nu<1 whose continued fraction is periodic, and the periodic sequence of terms is composed of a string of p-1 1s followed by an integer n>1,

 nu_(p,n)=[0,1,1,...,1_()_(p-1),n^_].
(1)

This can be written in the form

 nu_(p,n)=[0,1,1,...,1_()_(p-1),n,nu_(p,n)^(-1)],
(2)

which can be solved to give

 nu_(p,n)=1/2n(sqrt(1+4(nF_(p-1)+F_(p-2))/(n^2F_p))-1),
(3)

where F_n is a Fibonacci number.

Special cases include

nu_(p,2)=sqrt((F_(p+2))/(F_p))-1
(4)
mu_(p,3)=1/2(sqrt((4F_(2p))/(F_p^2)+9)-3).
(5)

See also

Periodic Continued Fraction, Noble Number

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References

Schroeder, M. R. Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 2nd enl. ed., corr. printing. Berlin: Springer-Verlag, 1990.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

Near Noble Number

Cite this as:

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

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