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# Paris Constant

The golden ratio can be written in terms of a nested radical in the beautiful form

 (1)

which can be written recursively as

 (2)

for , with .

Paris (1987) proved approaches at a constant rate, namely

 (3)

as , where

 (4)

(OEIS A105415) is the Paris constant.

A product formula for is given by

 (5)

(Finch 2003, p. 8).

Another formula is given by letting be the analytic solution to the functional equation

 (6)

for , subject to initial conditions and . Then

 (7)

(Finch 2003, p. 8).

A close approximation is , which is good to 4 decimal places (M. Stark, pers. comm.).

Portions of this entry contributed by Ed Pegg, Jr. (author's link)

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## References

Finch, S. R. "Analysis of a Radical Expansion." §1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.Paris, R. B. "An Asymptotic Approximation Connected with the Golden Number." Amer. Math. Monthly 94, 272-278, 1987.Plouffe, S. "The Paris Constant." http://pi.lacim.uqam.ca/piDATA/paris.txt.Sloane, N. J. A. Sequence A105415 in "The On-Line Encyclopedia of Integer Sequences."

Paris Constant

## Cite this as:

Pegg, Ed Jr. and Weisstein, Eric W. "Paris Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParisConstant.html