Talk:Tetration

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Tetration to infinite heights

The article should show the existence and the coordinates of the inflection point on the curve of y=x↑↑∞ within the domain interval [e⁻ᵉ , ᵉ√e]. It is somewhere near x=0.3944, y=0.5819, but greater precision should be used when presenting it. 50.110.99.89 (talk) 17:07, 18 December 2023 (UTC)[reply]

I used WolframAlpha to get those values at a much higher precision, so here it is:

x = 0.394416066798979621617062515433809095220247702822598132414641394633374582791277160634942

y = 0.581932705608592190120933697309893947888922734948425097472649678032890093766822476175890

I think that might be too much, but at least I helped you out with the precision. 107.9.41.132 (talk) 17:03, 5 October 2024 (UTC)[reply]

OEIS entry for 0.581932705608592190... —

Q

uantling (talk | contribs) 17:35, 6 October 2024 (UTC)[reply]

Tetration

What is 456 tetrated to 789? 2A02:C7C:5F3D:D500:7DA0:1FC2:12EF:D8B1 (talk) 21:14, 23 December 2023 (UTC)[reply]

456^^789 is congruent modulo 10^20 to 96042614856384249856. Moreover, 456^^789 is congruent modulo 10^790 to 456^^(789+c) for every positive integer c (the proof easily follows from my paper entitled "The congruence speed formula" (DOI: 10.7546/nntdm.2021.27.4.43-61)). --Marcokrt (talk) 14:29, 5 January 2024 (UTC)[reply]

Integer tetration peculiar property

In the "Properties" section of the Tetration page, I think that the constancy of the congruence speed should be mentioned since it is a peculiar property of hyper-4, it has been proven to hold (in radix-10, the well-known decimal numeral system) for any base that is not a multiple of 10 (see https://arxiv.org/pdf/2208.02622.pdf), and an explicit formula has also been given (see Equation 16 of https://nntdm.net/volume-28-2022/number-3/441-457/). Now, I am not going to edit the mentioned section since I received warnings in the past for this kind of stuff, but I will be glad to help you and provide proper references if someone thinks that such a result is worth mentioning. As a (trivial) special case, knowing that the constant congruence speed of the tetration base 3 is equal to 0 iff the hyperexponent is 1 and that it is 1 otherwise, we can state that Graham's number, G:=3^^b, is congruent modulo 10^(b-1) to 3^^c for any integer c=b+1,b+2,... and at the same time that the b-th rightmost digit of Graham's number is not the same of 3^^c for any integer c greater than b. Marcokrt (talk) 03:59, 5 January 2024 (UTC)[reply]

Added a short description (in parentheses) of a peculiar property characterizing integer tetration (i.e., tetration is the only hyperoperator having a constant congruence speed for nontrivial bases), providing a couple of references to the above-mentioned result since it is not easy to properly state it in less than a few lines.

In the above, I implicitly assumed radix-10, but it would be possible to derive analogous rules for any other square-free numeral system. Marcokrt (talk) 18:00, 7 January 2024 (UTC)[reply]

Unfortunately, somebody (anonimously) is trying to blank any contribution related to the discovery of the constancy of the congruence speed on the Web, deleting entire sections/pages with no reason (just as a personal attack against myself, I guess - see 16:18, 4 May 2024‎), and this occurred multiple times (see for instance https://wiki.riteme.site/w/index.php?title=Graham%27s_number&action=history - 16:23, 4 May 2024‎) on different platforms on the same couple of days, May 3rd and 4th, 2024, as you can see here (https://googology.fandom.com/wiki/Graham%27s_number?action=history - 16:43, 3 May 2024 ‎).

Now, I hope that this kind of vandalism can be prevented and forbidden in order to avoid losing the relevant information that these unknown people are trying to hide for some sort of personal reason and without providing any serious argument able to disprove peer-review results (and I am myself an endorser for the arXiv section math.NT who studyied this very specific topic for 10+ years, not some random amateur/mathematical crank... just to clarify the point). Thanks in advance for understanding!

Marco Marcokrt (talk) 22:28, 6 May 2024 (UTC)[reply]

The IP is correct, and was not commiting vandalism. This is a nonnotable result added by a person with a plain conflict of interest based on an unreliable journal. It never should have been added to the article in the first place. MrOllie (talk) 01:58, 7 May 2024 (UTC)[reply]

Rounding in the Examples table

I think the Examples table has an issue with spurious precision. Right now ${}^{6}2$ is stated to be 2.12004 × 10^{6.03123×10^19,727}. I don't think the 2.12004 factor should be there because the rounding of 6.03123 has a much larger effect on the accuracy of the number.

The number was calculated by finding the common log of ${}^{6}2$ . The integer part of the result starts with 60312260... and is 19,728 digits long. The fractional part starts with .3263437.... Then 10 is raised to the power of the integer and fractional parts to get the value of ${}^{6}2$ . The fractional part gives us $10^{0.3263437...}=2.120038...$ . This is where that factor comes from.

But remember that the integer part of the log is 19,728 digits long. It was rounded to 6.03123×10^19,727. Adding just one more digit of accuracy is a change of 4×10^19,721. The inaccuracy from this rounding vastly overshadows the 2.12 figure. It would be like saying the sun is 93 million miles and 4.2 inches away. If the 93 is rounded at all then the 4.2 is meaningless. Likewise, I believe the 2.12004 is meaningless.

Another way to calculate this would be as follows:

${}^{6}2=2^{{}^{5}2}\approx 2^{2.00353\times 10^{19,728}}=10^{\log _{10}{(2)}\times 2.00353\times 10^{19,728}}\approx 10^{0.603123\times 10^{19,728}}=10^{6.03123\times 10^{19,727}}$

I will be changing this as well as ${}^{4}4$ , ${}^{4}5$ , and ${}^{4}6$ for the same reason. It's quite possible that I made a mistake in my math or in my reasoning so if anyone wants to double check this it would be very helpful. Jak86 (talk)(contribs) 05:45, 23 April 2024 (UTC)[reply]

For

{}^{6}2

the first few digits are 21200, we're just arguing over how many millions or billions of digits follow. The rounded part is in the integral power that ten will be raised to: it changes where the digits are, but not what they are. The number that has been rounded will always just be a one followed by some number of zeros: rounding changes the number of zeros. Adding one more digit of accuracy to the exponent of ten changes the number dramatically, but changes 2.12004 in no way at all. This is the magic of logarithms, and fundamentally how floating point math works. 66.113.23.42 (talk) 23:38, 20 July 2024 (UTC)[reply]

Open question?

The article states “It is not known whether ⁿq is rational for any positive integer n and positive non-integer rational q”. This doesn’t make any sense, as (if I’m understanding the concept of titration correctly) ¹0.5 would be equal to 0.5, and would fit all 3 criteria:

n is a positive integer
q is a position non-integer rational
ⁿq is rational

Is the article supposed to say “every positive integer n”? 203.220.166.72 (talk) 05:31, 14 September 2024 (UTC)[reply]

Real heights

The section on real heights has serious issues. The continuity, differentiability, and regularity requirements do not seem to produce inconsistent results, contrary to the article (added a citation needed for this). In fact, [15], the Paulsen & Cowgill paper referenced in the complex heights section gives a result which fulfills all of these. This entire section should likely be reworked with this and subsequent papers in mind. 50.1.19.82 (talk) 19:42, 12 October 2024 (UTC)[reply]