Thread Rating:
  • 2 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
pentation and hexation
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. I think from this we can show that anything less than 2 does not grow unbounded, (just observe the meaning of the n'th hyperoperator root). Any base less than the n'th eta constant has a bounded hyperoperator for N>n-1.

We get something similar with the bounded hyperoperators as what you're talking about. It's a little messier but essentially there exists   n-2'th hyperoperators. They satisfy the following really weird recursion

for ANY arbitrary b, and fixed. The first n numbers have to equal for this recursion to work though, but the last number can be anything. Essentially, everytime we go up a hyperoperator, we get another amount of iterates.

I haven't actually proved this, but it's really straightforward.

Firstly, there exists exponentiations. namely for all integer a_1. Then we fix an a_1 and there exists a kind of fixed point that is unique (its uniqueness and what qualifies it is a little hard to explain, which is the weakest point of my argument). It's geometrically attracting and hence can be iterated. Now the iterate  is conjugate to SOME exponential for the multiplier of the above fixed point. This generates tetration functions g(z,1). (they interpolate the natural power towers, send to complex values and are bounded on SOME half plane of the complex plane that includes , and satisfy the recursion . This allows us to find another FIXED point and it's unique because of Schwarz' lemma. Rinse and repeat, and we get pentations. Then hexations, so on and so forth.

This is all really roughly argued, more intuition than rigor. But I call it the "branches of hyper operators."

Messages In This Thread
pentation and hexation - by sheldonison - 08/07/2017, 07:33 PM
RE: pentation and hexation - by JmsNxn - 08/21/2017, 08:05 PM
RE: pentation and hexation - by sheldonison - 08/22/2017, 02:03 PM
RE: pentation and hexation - by JmsNxn - 08/22/2017, 10:38 PM
RE: pentation and hexation - by sheldonison - 09/03/2017, 10:11 PM
RE: pentation and hexation - by JmsNxn - 09/03/2017, 11:52 PM
RE: pentation and hexation - by sheldonison - 09/04/2017, 03:04 AM
RE: pentation and hexation - by JmsNxn - 09/04/2017, 04:07 AM
RE: pentation and hexation - by Ember Edison - 09/18/2019, 06:34 AM
RE: pentation and hexation - by sheldonison - 09/18/2019, 02:34 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Tetration is pentation. This deserve more thinking. marraco 2 6,573 03/30/2015, 02:54 PM
Last Post: marraco
  [2015] 4th Zeration from base change pentation tommy1729 5 11,117 03/29/2015, 05:47 PM
Last Post: tommy1729
  Mizugadro, pentation, Book Kouznetsov 41 86,715 03/02/2015, 08:13 PM
Last Post: sheldonison
  Infinite Pentation (and x-srt-x) andydude 20 42,266 05/31/2011, 10:29 PM
Last Post: bo198214
  Regular "pentation"? mike3 12 32,143 04/04/2011, 03:16 AM
Last Post: BenStandeven
Smile Pentation roots self but please you do... nuninho1980 2 10,123 11/03/2010, 12:54 PM
Last Post: nuninho1980
  Pentation's definitional ambiguity Base-Acid Tetration 14 34,567 12/15/2009, 11:23 PM
Last Post: Base-Acid Tetration
  Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation Base-Acid Tetration 19 47,402 10/24/2009, 04:12 AM
Last Post: andydude
  Exploring Pentation - Base e jaydfox 22 55,156 03/03/2008, 08:04 PM
Last Post: Ivars

Users browsing this thread: 1 Guest(s)