Power of three
This article possibly contains original research. (April 2023) |
In
In a context where only integers are considered, n is restricted to non-negative values, so there are 1, 3, and 3 multiplied by itself a certain number of times.
The first ten powers of 3 for non-negative values of n are:
- )
Applications
The powers of three give the place values in the ternary numeral system.[1]
Graph theory
In
Enumerative combinatorics
In enumerative combinatorics, there are 3n signed subsets of a set of n elements. In polyhedral combinatorics, the hypercube and all other Hanner polytopes have a number of faces (not counting the empty set as a face) that is a power of three. For example, a 2-cube, or square, has 4 vertices, 4 edges and 1 face, and 4 + 4 + 1 = 32. Kalai's 3d conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope.[5]
Inverse power of three lengths
In
Perfect totient numbers
In number theory, all powers of three are perfect totient numbers.[10] The sums of distinct powers of three form a Stanley sequence, the lexicographically smallest sequence that does not contain an arithmetic progression of three elements.[11] A conjecture of Paul Erdős states that this sequence contains no powers of two other than 1, 4, and 256.[12]
Graham's number
Graham's number, an enormous number arising from a proof in Ramsey theory, is (in the version popularized by Martin Gardner) a power of three. However, the actual publication of the proof by Ronald Graham used a different number which is a power of two and much smaller.[13]
Table of values
(sequence A000244 in the OEIS)
30 | = | 1
|
316 | = | 43,046,721 | 332 | = | 1,853,020,188,851,841 | 348 | = | 79,766,443,076,872,509,863,361 | |||
31 | = | 3
|
317 | = | 129,140,163 | 333 | = | 5,559,060,566,555,523 | 349 | = | 239,299,329,230,617,529,590,083 | |||
32 | = | 9
|
318 | = | 387,420,489 | 334 | = | 16,677,181,699,666,569 | 350 | = | 717,897,987,691,852,588,770,249 | |||
33 | = | 27 | 319 | = | 1,162,261,467 | 335 | = | 50,031,545,098,999,707 | 351 | = | 2,153,693,963,075,557,766,310,747 | |||
34 | = | 81 | 320 | = | 3,486,784,401 | 336 | = | 150,094,635,296,999,121 | 352 | = | 6,461,081,889,226,673,298,932,241 | |||
35 | = | 243 | 321 | = | 10,460,353,203 | 337 | = | 450,283,905,890,997,363 | 353 | = | 19,383,245,667,680,019,896,796,723 | |||
36 | = | 729 | 322 | = | 31,381,059,609 | 338 | = | 1,350,851,717,672,992,089 | 354 | = | 58,149,737,003,040,059,690,390,169 | |||
37 | = | 2,187 | 323 | = | 94,143,178,827 | 339 | = | 4,052,555,153,018,976,267 | 355 | = | 174,449,211,009,120,179,071,170,507 | |||
38 | = | 6,561 | 324 | = | 282,429,536,481 | 340 | = | 12,157,665,459,056,928,801 | 356 | = | 523,347,633,027,360,537,213,511,521 | |||
39 | = | 19,683 | 325 | = | 847,288,609,443 | 341 | = | 36,472,996,377,170,786,403 | 357 | = | 1,570,042,899,082,081,611,640,534,563 | |||
310 | = | 59,049 | 326 | = | 2,541,865,828,329 | 342 | = | 109,418,989,131,512,359,209 | 358 | = | 4,710,128,697,246,244,834,921,603,689 | |||
311 | = | 177,147 | 327 | = | 7,625,597,484,987 | 343 | = | 328,256,967,394,537,077,627 | 359 | = | 14,130,386,091,738,734,504,764,811,067 | |||
312 | = | 531,441 | 328 | = | 22,876,792,454,961 | 344 | = | 984,770,902,183,611,232,881 | 360 | = | 42,391,158,275,216,203,514,294,433,201 | |||
313 | = | 1,594,323 | 329 | = | 68,630,377,364,883 | 345 | = | 2,954,312,706,550,833,698,643 | 361 | = | 127,173,474,825,648,610,542,883,299,603 | |||
314 | = | 4,782,969 | 330 | = | 205,891,132,094,649 | 346 | = | 8,862,938,119,652,501,095,929 | 362 | = | 381,520,424,476,945,831,628,649,898,809 | |||
315 | = | 14,348,907 | 331 | = | 617,673,396,283,947 | 347 | = | 26,588,814,358,957,503,287,787 | 363 | = | 1,144,561,273,430,837,494,885,949,696,427 |
All of these numbers above represent exponents in
Powers of three whose exponents are powers of three
(sequence A055777 in the OEIS)
31 | = | 3
|
1 digit |
33 | = | 27 | 2 digits |
39 | = | 19,683 | 5 digits |
327 | = | 7, |
13 digits |
381 | = | 443, |
39 digits |
3243 | = | 87, |
116 digits |
3729 | = | 662, |
347 digits |
32187 | = | 291, |
1,044 digits |
36561 | = | 24, |
3,131 digits |
319683 | = | 15, |
9,392 digits |
359049 | = | 3, |
28,174 digits |
3177147 | = | 39, |
84,521 digits |
All of these numbers above end in 3 or 7.[citation needed]
The numbers form an irrationality sequence: for every sequence of
converges to an irrational number. Despite the rapid growth of this sequence, it is a slow-growing irrationality sequence.[citation needed]
Selected powers of three
33 = 27
The number that is the
39 = 19,683
The largest power of three with distinct digits in
327 = 7,625,597,484,987
The number that is the first power of three tetration of three.
339 = 4,052,555,153,018,976,267
The first power of 3 to contain all decimal digits.
368 = 278,128,389,443,693,511,257,285,776,231,761
The number that is conjectured to be the last power of 3 not containing a 0 in decimal.
3209 = 5,228,080,143,043,843,084,895,232,761,630,250,394,879,802,048,576,763,864,267,558,971,910,557,498,410,330,867,878,474,031,283,071,683
The largest power of 3 smaller than a googol (10100).
3210 = 15,684,240,429,131,529,254,685,698,284,890,751,184,639,406,145,730,291,592,802,676,915,731,672,495,230,992,603,635,422,093,849,215,049
The smallest power of 3 greater than a googol (10100).
See also
- Graham's Number
- Power of 10
- Power of two
- Square root of 3
References
- JSTOR 41185884
- S2CID 9855414
- ^ For the Brouwer–Haemers and Games graphs, see Bondarenko, Andriy V.; Radchenko, Danylo V. (2013), "On a family of strongly regular graphs with ", MR 0782310
- S2CID 8917264
- JFM 35.0387.02
- MR 2076132
- MR 3026271
- MR 2051959
- ^ Sloane, N. J. A. (ed.), "Sequence A005836", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
- MR 0580438