e Approximations

An amazing pandigital approximation to  that is correct to 18457734525360901453873570 decimal digits is given by

(1)

found by R. Sabey in 2004 (Friedman 2004).

Castellanos (1988ab) gives several curious approximations to ,

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

which are good to 6, 7, 9, 10, 12, and 15 digits respectively.

E. Pegg Jr. (pers. comm., Mar. 2, 2002), found

(8)

which is good to 7 digits.

J. Lafont (pers. comm., MAy 16, 2008) found

(9)

where  is a harmonic number, which is good to 7 digits.

N. Davidson (pers. comm., Sept. 7, 2004) found

(10)

which is good to 6 digits.

D. Barron noticed the curious approximation

(11)

where  is Catalan's constant and  is the Euler-Mascheroni constant, which however, is only good to 3 digits.

REFERENCES:

Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.

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