OFFSET
0,1
COMMENTS
Expressions A^a = A*a for A > 1 have two real solutions, one of which is trivial a = 1 (the exception being A = e = 2.718... where both solutions coincide).
Infinite expressions for a can arise from Pi^a = Pi*a, such as, a = Pi^(Pi^(Pi^(...)-1)-1) and a = log_Pi(Pi*log_Pi(Pi*log_Pi(...))), however, they are by themselves not well defined, because the values of their partial expressions do not converge or converge to different values depending on how the expression is truncated.
a satisfies (1-log(Pi))(a-1) - (a-1)^2/2 + (a-1)^3/3 - (a-1)^4/4 + (a-1)^5/5 - ... = 0
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 digits from Rok Cestnik)
Rok Cestnik, Plot of the difference Pi^a-Pi*a
Eric Weisstein's World of Mathematics, Lambert W-Function
Wikipedia, Lambert W function
FORMULA
Equals -LambertW(-log(Pi)/Pi)/log(Pi). - Alois P. Heinz, Mar 14 2018
EXAMPLE
0.7582711543684640692301140455152168913364...
MATHEMATICA
NSolve[Pi^a == Pi*a, a, 100]
(* Second program: *)
RealDigits[-ProductLog[-Log[Pi]/Pi]/Log[Pi], 10, 100][[1]] (* Jean-François Alcover, Oct 06 2019 *)
PROG
(PARI) solve(x=0, 0.9, Pi^x - Pi*x) \\ Michel Marcus, Jan 08 2017
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Rok Cestnik, Jan 08 2017
STATUS
approved