Talk:Tetration: Difference between revisions
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* <nowiki>[[Limit (mathematics)#Limit of a sequence|convergence of the series]]</nowiki> Anchor [[Limit (mathematics)#Limit of a sequence]] links to a specific web page: [[Limit of a sequence]]. The anchor (#Limit of a sequence) has been [[Special:Diff/1106612212|deleted by other users]] before. <!-- {"title":"Limit of a sequence","appear":{"revid":349810035,"parentid":349809994,"timestamp":"2001-11-11T00:30:23Z","removed_section_titles":[],"added_section_titles":["Limit of a sequence","Examples:","Properties","Limit of a function","Examples:","Properties","Generalizations"]},"disappear":{"revid":1106612212,"parentid":1106477201,"timestamp":"2022-08-25T13:57:06Z","removed_section_titles":["Limit of a function","Limit of a sequence","Limit as \"standard part\"","Convergence and fixed point","Computability of the limit"],"added_section_titles":["Properties","Convergence","Computability","Types","In sequences","In functions","In nonstandard analysis"]}} --> |
* <nowiki>[[Limit (mathematics)#Limit of a sequence|convergence of the series]]</nowiki> Anchor [[Limit (mathematics)#Limit of a sequence]] links to a specific web page: [[Limit of a sequence]]. The anchor (#Limit of a sequence) has been [[Special:Diff/1106612212|deleted by other users]] before. <!-- {"title":"Limit of a sequence","appear":{"revid":349810035,"parentid":349809994,"timestamp":"2001-11-11T00:30:23Z","removed_section_titles":[],"added_section_titles":["Limit of a sequence","Examples:","Properties","Limit of a function","Examples:","Properties","Generalizations"]},"disappear":{"revid":1106612212,"parentid":1106477201,"timestamp":"2022-08-25T13:57:06Z","removed_section_titles":["Limit of a function","Limit of a sequence","Limit as \"standard part\"","Convergence and fixed point","Computability of the limit"],"added_section_titles":["Properties","Convergence","Computability","Types","In sequences","In functions","In nonstandard analysis"]}} --> |
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=log(-1)(-1),…,here,we also ought to define that log(-1)(-1)=-1,so (-1)↑↑0=log(-1)[(-1)↑↑1] |
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=log(-1)(-1)=-1≠1, |
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(-1)↑↑(-1)=log(-1)[(-1)↑↑0] |
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=log(-1)(-1)=-1≠0, |
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(-1)↑↑(-2)=log(-1)[(-1)↑↑(-1)] |
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=-1=log(-1)(-1)≠-∞. |
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(-1)↑↑(-3)=log(-1)[(-1)↑↑(-2)] |
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=-1=log(-1)(-1). |
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== Tetration to infinite heights == |
== Tetration to infinite heights == |
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Revision as of 01:33, 23 April 2024
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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)
Usage in speech
What's the correct way to refer to the tetration operation when reading a mathematical expression out loud? For example, when reading the expression 3↑↑5, would you say, "three tetrated to five," or "three to the fifth tetration," or some such thing? — Preceding unsigned comment added by Mvrog (talk • contribs) 23:09, 27 April 2023 (UTC)
Tetration
What is 456 tetrated to 789? 2A02:C7C:5F3D:D500:7DA0:1FC2:12EF:D8B1 (talk) 21:14, 23 December 2023 (UTC)
- 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)
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)
- 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)