In mathematics, a power of three is a number of the form 3n where n is an integer, that is, the result of exponentiation with number three as the base and integer n as the exponent.
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:
The powers of three give the place values in the ternary numeral system.[1]
In graph theory, powers of three appear in the Moon–Moser bound 3n/3 on the number of maximal independent sets of an n-vertex graph,[2] and in the time analysis of the Bron–Kerbosch algorithm for finding these sets.[3] Several important strongly regular graphs also have a number of vertices that is a power of three, including the Brouwer–Haemers graph (81 vertices), Berlekamp–van Lint–Seidel graph (243 vertices), and Games graph (729 vertices).[4]
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]
In recreational mathematics and fractal geometry, inverse power-of-three lengths occur in the constructions leading to the Koch snowflake,[6] Cantor set,[7] Sierpinski carpet and Menger sponge, in the number of elements in the construction steps for a Sierpinski triangle, and in many formulas related to these sets. There are 3n possible states in an n-disk Tower of Hanoi puzzle or vertices in its associated Hanoi graph.[8] In a balance puzzle with w weighing steps, there are 3w possible outcomes (sequences where the scale tilts left or right or stays balanced); powers of three often arise in the solutions to these puzzles, and it has been suggested that (for similar reasons) the powers of three would make an ideal system of coins.[9]
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, 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]
(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 base-3, as mentioned above.
(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 positive integers, the series
converges to an irrational number. Despite the rapid growth of this sequence, it is a slow-growing irrationality sequence.[citation needed]
33 = 27
The number that is the cube of three.
39 = 19,683
The largest power of three with distinct digits in base-10.
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).