09/04/2017, 03:04 AM
(This post was last modified: 09/04/2017, 02:54 PM by sheldonison.)

(09/03/2017, 11:52 PM)JmsNxn Wrote: There is a result posted on here about how the "eta constants" converge to 2 and the "euler constants" converge to 4. The n'th eta constant is the sup of the x'th n'th hyperoperator root of x. And the "euler constants" are the actual values x_n such that x_n'th n'th hyperoperator root of x_n = n'th eta constant...

How interesting! I haven't seen this conjecture before. This is base 1.84, for which septation almost has an upper fixed point, that would be somewhere near 3.7 So octation for base=1.84 wouldn't quite be bounded, but oct(45)~=4.05, so it would take awhile before octation escapes; I can only calculate integer values for octation, until I do a theta mapping. Also, notice that the solution for base(e) above, behaves nicer than base 1.84.

Why would do the "euler constants" converge to 4, as n gets arbitrarily large for the "nth" hyperoperator for bases approaching 2?

- Sheldon