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%I #83 Sep 22 2023 05:17:52
%S 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,
%T 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,
%U 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
%N Decimal expansion of exp(Pi).
%C 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.
%C 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
%C e^Pi is transcendental, as proved by Gelfond. - _Charles R Greathouse IV_, May 07 2013
%C Nesterenko proves that this constant is algebraically independent of Pi and Gamma(1/4) over Q. - _Charles R Greathouse IV_, Nov 11 2013
%C Sum of the volumes of all even-dimensional unit spheres. - _Paolo Xausa_, Nov 14 2021
%D L. Berggren, J. Borwein and P. Borwein, "Pi: a source Book", second edition, Springer, p. 422
%H Harry J. Smith, <a href="/A039661/b039661.txt">Table of n, a(n) for n = 2..20000</a>
%H Bikash Chakraborty, <a href="https://arxiv.org/abs/1806.03163">A Visual Proof that Pi^e < e^Pi</a>, arXiv:1806.03163 [math.HO], 2018.
%H D. Hilbert, <a href="http://dx.doi.org/10.1090/S0273-0979-00-00881-8">Mathematical Problems</a>, Bull. Amer. Math. Soc. 37 (2000), 407-436. Reprinted from Bull. Amer. Math. Soc. 8 (Jul 1902), 437-479. See Problem 7.
%H Fouad Nakhli, <a href="https://www.jstor.org/stable/2689563">Proof without Words Pi^e < e^Pi</a>, Mathematics Magazine, 60(3) (1987), pp. 165.
%H Yu V. Nesterenko, <a href="http://dx.doi.org/10.1070/SM1996v187n09ABEH000158">Modular functions and transcendence questions</a>, Sbornik: Mathematics 187:9 (1996), pp. 1319-1348. (English translation)
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap28.html">exp(pi) to 5000 digits</a>
%H Arjun K. Rathie, Gradimir V. Milovanović, and Richard B. Paris, <a href="http://www.mi.sanu.ac.rs/~gvm/radovi/AKR-GVM-RBP-MV.pdf">Hypergeometric representations of Gelfond's constant and its generalisations</a>, Serbian Academy of Sciences and Arts (2021).
%H H. S. Uhler, <a href="http://www.jstor.org/stable/2972387">On the numerical value of i^i</a>, Amer. Math. Monthly, 28 (1921), 114-116.
%H Andrés Vallejo and Italo Bove, <a href="https://arxiv.org/abs/2309.10826">Which is greater: e^Pi or Pi^e? An unorthodox solution to a classic puzzle</a>, arXiv:2309.10826 [physics.class-ph], 2023.
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/GelfondsConstant.html">Gelfond's Constant</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gelfond's_constant">Gelfond's constant</a>
%H OEIS Wiki, <a href="/wiki/Gelfond's_constant">Gelfond's constant</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F 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
%F 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
%F exp(Pi) ~= log(Pi) + 7*Pi. - _Alexander R. Povolotsky_, Oct 24 2009
%F Equals Sum_{k>=0} Pi^k/k!. - _Paolo Xausa_, Nov 14 2021
%e 23.1406926327792690...
%t RealDigits[N[E^Pi,200]] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2010 *)
%o (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
%o (PARI) A039661(n)=default(realprecision,n);exp(Pi)\10^(3-n)%10 \\ _M. F. Hasler_, Oct 24 2009
%Y Cf. A059850 (Pi^e).
%Y Cf. A058287 = contfrac(e^Pi), A058288 = contfrac(Pi^e).
%K nonn,cons
%O 2,1
%A _N. J. A. Sloane_
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