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Phi Number System

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The phi number system represents a positive integer n by a unique finite set of distinct nonconsecutive (not necessarily positive) integer exponents {k_1,...,k_m} such that

n=phi^(k_1)+...+phi^(k_m),
(1)

where phi is the golden ratio.

For example, for the first few positive integers,

1=phi^0
(2)
2=phi^(-2)+phi
(3)
3=phi^(-2)+phi^2
(4)
4=phi^(-2)+phi^0+phi^2
(5)
5=phi^(-4)+phi^(-1)+phi^3
(6)
6=phi^(-4)+phi+phi^3
(7)
7=phi^(-4)+phi^4
(8)

(OEIS A104605).

The numbers of terms needed to represent n for n=1, 2, ... are given by 1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 4, ... (OEIS A055778), which are also the numbers of 1s in the base-phi representation of n.

The following table summarizes the values of n that require exactly k powers of phi in their representations.

kOEISnumbers requiring exactly k powers
2A0052482, 3, 7, 18, 47, 123, 322, 843, ...
3A1046264, 5, 6, 8, 19, 48, 124, 323, 844, ...
4A1046279, 10, 12, 13, 14, 16, 17, 20, 21, 25, ...
5A10462811, 15, 22, 23, 24, 26, 30, 31, 32, 34, ...

See also

Base, Golden Ratio

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References

Bergman, G. "A Number System with an Irrational Base." Math. Mag. 31, 98-110, 1957.Knott, R. "Using Powers of Phi to represent Integers (Base Phi)." http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phigits.html.Knuth, D. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1997.Levasseur, K. "The Phi Number System." http://www.hostsrv.com/webmaa/app1/MSP/webm1010/PhiNumberSystem/PhiNumberSystem.msp.Rousseau, C. "The Phi Number System Revisited." Math. Mag. 68, 283-284, 1995.Sloane, N. J. A. Sequences A005248/M0848, A055778, A104605, A104626, A104627, and A104628 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Phi Number System

Cite this as:

Weisstein, Eric W. "Phi Number System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PhiNumberSystem.html

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