2.86295 + 3.22327 i

[tommy1729]

Note I discuss \( ^na=1 \); Tetration research: 1986 - 1991, pg 4.

Yes that is related !

z^^2 = 1 has the solution 2.2136 + 3.1140 i.

I was thinking about solving z^^3 = z today.

z^^3 = z^(z^^2) = z^1 = z.
so 2.2136 + 3.1140 i is a solution.


z^^3 = z = z^(z^z) 
so if z^z = z then z^(z^z) = z^z = z.

therefore 2.86295 + 3.22327 i from the OP is also a solution.

another solution we get from

z^^3 = z
=>
exp( ln(z) * z^z ) = exp(ln(z))
=>
ln(z) * z^z = ln(z) + 2pi i
so z = 2.36678 + 0.617735 i

***

I suspect that z^z^z = z has "2 or 3 rays" of solutions.
whereas z^z = z only has 1 ray.
Look at gottfried's post of the zero's ; the zero's are on a nice curve ... that's what i call a "ray".
informally : Ray's are more or less 1 dimensional set of points as opposed to say the gaussian integers.



In general I think z^^n = z has about n rays of solutions.
I added the informal definition and counterexample to make the statement somewhat falsifiable.

I like that when we pick the smallest element of the rays it is almost like z^^n = z is like solving a polynomial of degree O(n). ( big O notation)

***

Inverting z^^n for fixed n is an interesting function but I have not seen much research for series expansions based upon their fixpoints ?
Maybe Im wrong or maybe that is not interesting ??

In the open problems section there is a conjectured series expansion for it btw !!

***

consider making a list of solutions to z^^k = z for rising index k.

As showed above we meet old solutions (small k) again for larger k.

And we can repeat it.

This is some elementary number theory :

if k-1 factors into k-1 = a*b then the solutions of index a and b are solutions too.

and then a - 1 = c*d etc.

But beware of multiplicities.

Now that is not unique to tetration ofcourse.
Any interesting number theory on this is appreciated !! even experimental !

... it is interesting to understand how this relates to the rays or the rays conjecture ...
if k-1 is primes do the rays shift a bit compared to those for k + a new ray or ??
what if k-1 mod g = small ??

Lots of questions.

Not even mentioned chaos.

***

As sheldon mentioned and as to be expected for chaos and fast speeds ; usually we jump from large to small values and vice versa by iterating.

But some numbers apparantly do not.

such as the fixpoints of order k.

Or limit cycles.

But even limit cycles jump to high and low values.
but not all.

I considered low values and long psuedoperiods with respect to the imput ( size of the base in normed value )

and I came fascinated by the simple base

5 + 9i.

5 + 9i makes nice patterns imho... or absolutely not depending on your view of nice.

That is informal but still.

some gaussian integers make nicer iterations then others and I lack deep understanding of it.


I FELT LIKE SAYING THOSE THINGS BECAUSE THEY MIGHT NOT BE OBVIOUS FROM NUMERICAL METHODS AND PLOTS !!

regards

tommy1729