Talk:Power of three

Improvements

If this article is to be kept (which I'm still against), we need to do the following:

1. Consider replacing 3^3^n by 3^2^n, and trimming the sequence or shorten the answers, as is done in Power of two#Powers of two whose exponents are powers of two. There's no justification given for including the section, as there is in "Power of two".
2. Trim the main Power of three#List of powers of three to 4 columns, and standardize commas. 4 columns here is a little wider than 6 columns in powers of 2, as ${\frac {\ln 3}{\ln 2}}\approx 1.58>1.5={\frac {6}{4}}.$
3. Source the "Selected powers of 3" section.

— Arthur Rubin (talk) 08:11, 8 March 2020 (UTC)

The largest power of three found

Can you add a note below 15,684,2...,215,049 for the largest power of three you can find? Faster328 (talk) 06:23, 10 April 2023 (UTC)

This seems unhelpful, as anyone can improve on the "largest" power simply by multiplying it by three. Certes (talk) 16:43, 10 April 2023 (UTC)

I think that you can add some powers of three whose exponents are powers of three, maybe 16. Faster328 (talk) 06:14, 11 April 2023 (UTC)

How would that improve the article? Is there any significant theorem or practical application for powers of three whose exponents are powers of three that doesn't apply to other powers of three? Certes (talk) 14:20, 11 April 2023 (UTC)

3 cubed is 27, 27 cubed is 19,683, 19,683 cubed is 7,625,597,484,987, 7,625,597,484,987 cubed is 443,426,488,243,037,769,948,249,630,619,149,892,803, 443,426,488,243,037,769,948,249,630,619,149,892,803 cubed is 87,189,642,485,960,958,202,911,070,585,860,771,696,964,072,404,731,750,085,525,219,437,990,967,093,723,439,943,475,549,906,831,683,116,791,055,225,665,627, and so on. Note that there is no 3^3^n that ends in 9 or 1. They follow a cycle which is 87-03-27-83-, which is, every tetration of three, will converge to a 10-adic number (...5,627,262,464,195,387). Faster328 (talk) 06:15, 12 April 2023 (UTC)

I already added the new section. Go check it out. It is controversial for 3^3^n to form an irrationality sequence. If this is true, keep it. Faster328 (talk) 07:44, 12 April 2023 (UTC)

Powers of three whose exponents are powers of three

(sequence A055777 in the OEIS)

31	=	3
33	=	27
39	=	19,683
327	=	7,625,597,484,987 (13 digits)
381	=	443,426,488,243,037,769,948,249,630,619,149,892,803 (39 digits)
3243	=	87,189,642,485,960,958,202,911,070,585,860,771,696,964,072,404,731,750,085,525,219,437,990,967,093,723,439,943,475,549,906,831,683,116,791,055,225,665,627 (116 digits)
3729	=	662,818,605,424,187,176,105,172,8...0,205,437,212,700,131,838,846,883 (347 digits)
32187	=	291,195,106,143,185,347,895,545,4...2,152,086,037,206,066,369 147,387 (1,044 digits)
36561	=	24,691,769,589,333,631,072,790,02...3,504,694,982,343,979,438,089,603 (3,131 digits)
319683	=	15,054,164,145,220,926,243,143,29...7,800,510,818,762,686,617,859,227 (9,392 digits)
359049	=	3,411,692,975,886,675,012,755,349,...1,240,649,572,770,556,941,930,083 (28,174 digits)
3177147	=	39,710,908,604,467,909,151,990,52...6,219,060,890,796,418,967,881,787 (84,521 digits)

All of these numbers above end in 3 or 7.

The numbers $3^{3^{n}}$ form an irrationality sequence: for every sequence $x_{i}$ of positive integers, the series

$\sum _{i=0}^{\infty }{\frac {1}{3^{3^{i}}x_{i}}}={\frac {1}{3x_{0}}}+{\frac {1}{27x_{1}}}+{\frac {1}{19683x_{2}}}+\cdots$

converges to an irrational number. Despite the rapid growth of this sequence, it is a slow-growing irrationality sequence. Faster328 (talk) 07:34, 12 April 2023 (UTC)

I already added this in the main article page. You need to find a citation for each (need for citation). And please do not expand or trim the table. Just right is 12 to 16 entries. Faster328 (talk) 07:38, 12 April 2023 (UTC)

I added 12 entries to the new table. Also, I added the controversy of 3^3^n forming an irrationality sequence. Faster328 (talk) 07:41, 12 April 2023 (UTC)

Consistency with 'Power of two' article

I wanted this article to be consistent with the 'Power of two' article. Faster328 (talk) 08:38, 12 April 2023 (UTC)

Negative integer exponents

Is there anything in sources about negative exponents, it numbers 1/3, 1/9, 1/27...? Or shal the lede speak about nonnegative integer exponents? Is there a ref for the def? (I mean, to establish that usually nonnegative exp is meant (that is my impression, but {{citation needed}} for either way :-). - Altenmann >talk 20:41, 29 March 2024 (UTC)