06/09/2017, 01:11 PM
(This post was last modified: 06/09/2017, 02:53 PM by sheldonison.)
(06/08/2017, 09:20 PM)JmsNxn Wrote: Really? Does the same happen for pentation? Does it have a real fixed point? Has anybody bothered to go around and actually construct \( e \uparrow^n x \) using Kneser's tetration as a base? I mean, if tetration has a repelling real fixed point then it's easy to get to pentation. If pentation has the same thing, it's easy to get hexation...Just a quick overview on pentation for bases>eta. I have not spent any time thinking about hexation for these bases. How far have you gotten in thinking about pentation and hexation for bases<eta?
This is a graph of Tet( 1.63532449671528 ) Tetration bases greater than eta, and less than this base have three real valued fixed points. Tetration bases greater than this base have one real fixed point. The most straightforward way to generate pentation is from the lower fixed point, somewhere between -2 and -1, using that real valued repelling fixed point. Nuinho first pointed out this Tetration base, with a parabolic upper fixed point.
And here is a graph of pentation base(2), from Tet_2's fixed point of -1.743909176132. This pentation base2 graph does not have any real valued fixed points. There are singularities in Pent(z+1) wherever Pent(z) is a negative integer<=-2; this should happen in the complex plane arbitrarily close to the real axis, as z gets bigger.
- Sheldon