Feigenbaum Constant Approximations

A curious approximation to the Feigenbaum constant  is given by

(1)

where  is Gelfond's constant, which is good to 6 digits to the right of the decimal point.

M. Trott (pers. comm., May 6, 2008) noted

(2)

where  is Gauss's constant, which is good to 4 decimal digits, and

(3)

where  is the tetranacci constant, which is good to 3 decimal digits.

A strange approximation good to five digits is given by the solution to

(4)

which is

(5)

where  is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).

(6)

gives  to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).

M. Hudson (pers. comm., Nov. 20, 2004) gave

			(7)
			(8)
			(9)

which are good to 17, 13, and 9 digits respectively.

Stoschek gave the strange approximation

(10)

which is good to 9 digits.

R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations

			(11)
			(12)
			(13)
			(14)
			(15)
			(16)

where e is the base of the natural logarithm and  is Gelfond's constant, which are good to 3, 3, 5, 7, 9, and 10 decimal digits, respectively, and

			(17)
			(18)
			(19)
			(20)
			(21)
			(22)
			(23)

which are good to 3, 3, 3, 4, 6, 8, and 8 decimal digits, respectively.

An approximation to  due to R. Phillips (pers. comm., Jan. 27, 2005) is obtained by numerically solving

(24)

for , where  is the golden ratio, which is good to 4 digits.

REFERENCES:

Friedman, E. "Problem of the Month (August 2004)." http://www.stetson.edu/~efriedma/mathmagic/0804.html.