A039661 Decimal expansion of exp(Pi).

2, 3, 1, 4, 0, 6, 9, 2, 6, 3, 2, 7, 7, 9, 2, 6, 9, 0, 0, 5, 7, 2, 9, 0, 8, 6, 3, 6, 7, 9, 4, 8, 5, 4, 7, 3, 8, 0, 2, 6, 6, 1, 0, 6, 2, 4, 2, 6, 0, 0, 2, 1, 1, 9, 9, 3, 4, 4, 5, 0, 4, 6, 4, 0, 9, 5, 2, 4, 3, 4, 2, 3, 5, 0, 6, 9, 0, 4, 5, 2, 7, 8, 3, 5, 1, 6, 9, 7, 1, 9, 9, 7, 0, 6, 7, 5, 4, 9, 2

Offset

2,1

Comments

e^Pi and Pi^e (A059850) differ by hardly 3% in magnitude. The determination of the inequality sign between them does not require their actual evaluation, the result being immediate from the basic facts Pi>e and log(x+1)<x for positive x, whence setting x=(Pi/e)-1 (>0) yields log(Pi)<Pi/e, or Pi^e < e^Pi.

The formulas give e^Pi, not a(n). Note that e^Pi - Pi = 19.999099979...; that's why e^Pi and 20 + Pi have many common decimal digits. - M. F. Hasler, Oct 24 2009

e^Pi is transcendental, as proved by Gelfond. - Charles R Greathouse IV, May 07 2013

Nesterenko proves that this constant is algebraically independent of Pi and Gamma(1/4) over Q. - Charles R Greathouse IV, Nov 11 2013

Sum of the volumes of all even-dimensional unit spheres. - Paolo Xausa, Nov 14 2021

References

L. Berggren, J. Borwein and P. Borwein, "Pi: a source Book", second edition, Springer, p. 422

Links

Harry J. Smith, Table of n, a(n) for n = 2..20000

Bikash Chakraborty, A Visual Proof that Pi^e < e^Pi, arXiv:1806.03163 [math.HO], 2018.

D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 37 (2000), 407-436. Reprinted from Bull. Amer. Math. Soc. 8 (Jul 1902), 437-479. See Problem 7.

Fouad Nakhli, Proof without Words Pi^e < e^Pi, Mathematics Magazine, 60(3) (1987), pp. 165.

Yu V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics 187:9 (1996), pp. 1319-1348. (English translation)

Simon Plouffe, exp(pi) to 5000 digits

Arjun K. Rathie, Gradimir V. Milovanović, and Richard B. Paris, Hypergeometric representations of Gelfond's constant and its generalisations, Serbian Academy of Sciences and Arts (2021).

H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.

Andrés Vallejo and Italo Bove, Which is greater: e^Pi or Pi^e? An unorthodox solution to a classic puzzle, arXiv:2309.10826 [physics.class-ph], 2023.

Eric Weisstein, Gelfond's Constant

Wikipedia, Gelfond's constant

OEIS Wiki, Gelfond's constant

Index entries for transcendental numbers

Formula

e^Pi = 32*Product_{j>=0} (u(j+1)/u(j))^2^(-j+1)) where u(0)=1 and v(0)=1/sqrt(2); u(n+1) = u(n)/2 + v(n)/2, v(n+1) = sqrt(u(n)v(n)) (deduced from Salamin algorithm for Pi). - Benoit Cloitre, Aug 14 2003

e^Pi = Sum_{k>=0} a(k)/k!/2^k where a(0)=1, a(1)=6 and a(n) = (40 - 4*n + n^2)*a(n-2) for n>=2 (from expansion of exp(6*asin(x)) at x=1/2). - Jaume Oliver Lafont, Oct 21 2009

exp(Pi) ~= log(Pi) + 7*Pi. - Alexander R. Povolotsky, Oct 24 2009

Equals Sum_{k>=0} Pi^k/k!. - Paolo Xausa, Nov 14 2021

Example

23.1406926327792690...

Mathematica

RealDigits[N[E^Pi, 200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

Prog

(PARI) default(realprecision, 20080); x=exp(1)^Pi/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b039661.txt", n, " ", d)); \\ Harry J. Smith, Apr 18 2009
(PARI) A039661(n)=default(realprecision, n); exp(Pi)\10^(3-n)%10 \\ M. F. Hasler, Oct 24 2009

Crossrefs

Cf. A059850 (Pi^e).

Cf. A058287 = contfrac(e^Pi), A058288 = contfrac(Pi^e).

Sequence in context: A306646 A152832 A211343 * A293668 A214684 A268727

Adjacent sequences: A039658 A039659 A039660 * A039662 A039663 A039664

Keyword

nonn,cons

Author

N. J. A. Sloane

Status

approved