e Approximations

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


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

Castellanos (1988ab) gives several curious approximations to e,

e approx 2+(54^2+41^2)/(80^2)
 approx (pi^4+pi^5)^(1/6)
 approx (271801)/(99990)
 approx (150-(87^3+12^5)/(83^3))^(1/5)
 approx 4-(300^4-100^4-1291^2+9^2)/(91^5)
 approx (1097-(55^5+311^3-11^3)/(68^5))^(1/7),

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

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

 e approx 3-sqrt(5/(63)),

which is good to 7 digits.

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

 e approx H_8(1+1/(80^2)),

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

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

 e approx 163^(32/163),

which is good to 6 digits.

D. Barron noticed the curious approximation

 e approx K^(gamma-5/7)pi^(gamma+2/7),

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

See also

Almost Integer, e

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Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.Friedman, E. "Problem of the Month (August 2004)."

Referenced on Wolfram|Alpha

e Approximations

Cite this as:

Weisstein, Eric W. "e Approximations." From MathWorld--A Wolfram Web Resource.

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