e Approximations
An amazing zeroless pandigital approximation to e that is correct to 18457734525360901453873570 decimal digits is given by
which was found by R. Sabey in 2004 (Friedman 2004). An improved zeroless pandigital approximation
which is correct to 8368428989068425943817590916445001887164 decimal digits, was found by D. Bamberger (pers. comm., Mar. 13, 2024; Friedman 2004 updated page). A pandigital approximation including 0
which is correct to 5447761679516886279045570843725804037563002422 decimal digits, was subsequently found by Reddit user Fastfaxr (2024).
Castellanos (1988ab) gives several curious approximations to e,
which are good to 6, 7, 9, 10, 12, and 15 digits respectively.
E. Pegg Jr. (pers. comm., Mar. 2, 2002), found
| [画像: e approx 3-sqrt(5/(63)), ] |
(10)
|
which is good to 7 digits.
J. Lafont (pers. comm., MAy 16, 2008) found
| [画像: e approx H_8(1+1/(80^2)), ] |
(11)
|
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), |
(12)
|
which is good to 6 digits.
D. Barron noticed the curious approximation
| e approx K^(gamma-5/7)pi^(gamma+2/7), |
(13)
|
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, eExplore with Wolfram|Alpha
More things to try:
References
Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.Fastfaxr. "New Pandigital Formula for e." Feb. 2025. https://www.reddit.com/r/math/comments/1j568wk/comment/nb0t2pw/.Friedman, E. "Problem of the Month (August 2004)." https://erich-friedman.github.io/mathmagic/0804.html.Referenced on Wolfram|Alpha
e ApproximationsCite this as:
Weisstein, Eric W. "e Approximations." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/eApproximations.html