OFFSET
2,2
COMMENTS
Given z > 0, there exist positive real numbers x < y, with x^y = y^x = z, if and only if z > e^e. In that case, 1 < x < e < y and (x, y) = ((1 + 1/t)^t, (1 + 1/t)^(t+1)) for some t > 0. (For example, t = 1 gives 2^4 = 4^2 = 16 > e^e.) Proofs of these classical results and applications of them are in Marques and Sondow (2010).
e^e = lim_{n->infinity} ((n+1)/n)^((n+1)^(n+1)/n^n), n > 0 an integer; cf. [Vernescu] wherein it is also stated that the assertions of the previous comment above were proved by Alexandru Lupas in 2006. - L. Edson Jeffery, Sep 18 2012
A weak form of Schanuel's Conjecture implies that e^e is transcendental--see Marques and Sondow (2012).
LINKS
Harry J. Smith, Table of n, a(n) for n = 2..20000
D. Marques and J. Sondow, The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture, arXiv:1212.6931 [math.NT], 2012-2013.
Simon Plouffe, exp(E) to 2000 places
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.
A. Vernescu, About the use of a result of Professor Alexandru Lupas to obtain some properties in the theory of the number e, Gen. Math., Vol. 15, No. 1 (2007), 75-80.
FORMULA
Equals Sum_{n>=0} e^n/n!. - Richard R. Forberg, Dec 29 2013
Equals Product_{n>=0} e^(1/n!). - Amiram Eldar, Jun 29 2020
EXAMPLE
15.15426224147926418976043027262991190552854853685613976914...
MATHEMATICA
RealDigits[ E^E, 10, 110] [[1]]
PROG
(PARI) exp(exp(1))
(PARI) { default(realprecision, 20080); x=exp(1)^exp(1)/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b073226.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009
(Magma) Exp(Exp(1)); // G. C. Greubel, May 29 2018
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Rick L. Shepherd, Jul 21 2002
STATUS
approved