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RE: Real and complex tetration - marcokrt - 04/03/2023
(04/03/2023, 07:46 PM)tommy1729 Wrote:(04/03/2023, 01:58 AM)marcokrt Wrote: A naive reply to Daniel's original question: also integer tetration depends in a certain way on complex tetration.
In order to try to explain my point, let us just take a look at Corollary 3.3./Equation (26) of my paper https://arxiv.org/pdf/2208.02622.pdf .
Basically (by assuming the standard decimal numeral system), it shows that if the tetration base is congruent to \( 5\pmod{10} \), then a peculiar property of integer tetration (i.e., the constancy of its congruence speed, which in the present case is guaranteed by the constraint that the height of the power tower is at least equal to three) uniquely describes a subset of those bases ending with the digit \( 5 \) returned by \( 90 \) degrees rotations on the complex plane (of course, we can achieve the same goal by using goniometric functions as showed by Equation (21)).
In the mentioned paper, in order to provide the inverse map of all the tetration bases with any given congruence speed, I invoked decadic integers and we know that \( \mathbb{Z}_{10} \) has the countability of the continuum, while this is not true for \( \mathbb{Z} \) and for \( \mathbb{Q} \).
Thus, I think that, if we wish to clearly see the whole picture and understand its true meaning (i.e., congruence speed \( \rightarrow \) phase displacement/"sfasamento" involving the rightmost unfrozen digits comparison \( \rightarrow \) chaos theory related stuff), including intrinsic properties characterizing hyper-4 itself (not only holomorphic functions constructed by us following tetration rules that we have previously defined, I mean), we need to look at \( \mathbb{C} \) before turning again our eyes on the "real" axis to the ground.
Talking about mod tetration, what do you think about this ?
https://math.eretrandre.org/tetrationforum/showthread.php?tid=1735
regards
tommy1729
I think that it is an interesting observation and it is maybe a consequence of the outcome that I resumed at the end of the proof of Lemma 1 (p. 247) of NNTDM, 26(3), pp. 245-260, where we just invoke Theorem 1 of Jolly's preprint Constructing the Primitive Roots of Prime Powers in order to show that the congruence speed of the base \(3\) is unitary and constant for any power tower whose heigth is \(\geq 2 \).
Hensel's (lifting) lemma is always a useful heuristic tool, but I would like to see if Germain's extension of the Spanish approach (see On the Equation a^x ≡ x (mod b)) is enough to prove your conjecture.
RE: Real and complex tetration - tommy1729 - 04/03/2023
what do you mean by " my conjecture " ?
that there are infinitely many ?
regards
tommy1729
RE: Real and complex tetration - marcokrt - 04/03/2023
(04/03/2023, 11:28 PM)tommy1729 Wrote: what do you mean by " my conjecture " ?
that there are infinitely many ?
regards
tommy1729
I meant that the sentence "Mick posted my idea of tetration generating all residues mod p" suggests a conjecture, but maybe I was wrong about its attribution, since the MSE thread tetration primitive root q mod p is by Mick himself.
RE: Real and complex tetration - JmsNxn - 04/04/2023
Absolutely fascinating, marko! Keep posting your papers here; I find this stuff utterly mind boggling. I was never that good at number theory (but I have a knack for analytic number theory)--so it's awesome to see this strange stuff you're doing. Even if a bunch of it goes over my head, lol.