Talk:Tetration

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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] =log(-1)(-1)=-1≠1, (-1)↑↑(-1)=log(-1)[(-1)↑↑0] =log(-1)(-1)=-1≠0, (-1)↑↑(-2)=log(-1)[(-1)↑↑(-1)] =-1=log(-1)(-1)≠-∞. (-1)↑↑(-3)=log(-1)[(-1)↑↑(-2)] =-1=log(-1)(-1).

Tetration to infinite heights[edit]

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]

Usage in speech[edit]

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 (talkcontribs) 23:09, 27 April 2023 (UTC)[reply]

Tetration[edit]

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[edit]

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]