Power of three

In mathematics, a power of three is a number of the form $3 n$ where $n$ is an integer, that is, the result of exponentiation with number three as the base and integer $n$ as the exponent.

81 (3⁴) combinations of weights of 1 (3⁰), 3 (3¹), 9 (3²) and 27 (3³) kg – each weight on the left pan, right pan or unused – allow integer weights from −40 to +40 kg to be balanced; the figure shows the positive values

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:

1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ... (sequence A000244 in the OEIS)

Applications

The powers of three give the place values in the ternary numeral system.[1]

Graph theory

In graph theory, powers of three appear in the Moon–Moser bound $3 n /3$ 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]

Enumerative combinatorics

In enumerative combinatorics, there are $3 n$ 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 $4 + 4 + 1 = 3 2$ . Kalai's $3 d$ conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope.[5]

Inverse power of three lengths

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 $3 n$ 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 $3 w$ 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]

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.[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 base-3, as mentioned above.

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	=	7,625,597,484,987	13 digits
3⁸¹	=	443,426,488,243,037,769,948,249,630,619,149,892,803	39 digits
3²⁴³	=	87,189,642,485,960,958,202,911,0...,831,683,116,791,055,225,665,627	116 digits
3⁷²⁹	=	662,818,605,424,187,176,105,172,...,205,437,212,700,131,838,846,883	347 digits
3²¹⁸⁷	=	291,195,106,143,185,347,895,545,...,152,086,037,206,066,369 147,387	1,044 digits
3⁶⁵⁶¹	=	24,691,769,589,333,631,072,790,0...,504,694,982,343,979,438,089,603	3,131 digits
3¹⁹⁶⁸³	=	15,054,164,145,220,926,243,143,2...,800,510,818,762,686,617,859,227	9,392 digits
3⁵⁹⁰⁴⁹	=	3,411,692,975,886,675,012,755,34...,240,649,572,770,556,941,930,083	28,174 digits
3¹⁷⁷¹⁴⁷	=	39,710,908,604,467,909,151,990,5...,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.

Selected powers of three

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 = 15,684,240,429,131,529,254,685,6...,992,603,635,422,093,849,215,049

The smallest power of 3 greater than a googol (10¹⁰⁰).

References

Ranucci, Ernest R. (December 1968), "Tantalizing ternary", The Arithmetic Teacher, 15 (8): 718–722, doi:10.5951/AT.15.8.0718, JSTOR 41185884
Moon, J. W.; Moser, L. (1965), "On cliques in graphs", Israel Journal of Mathematics, 3: 23–28, doi:10.1007/BF02760024, MR 0182577, S2CID 9855414
Tomita, Etsuji; Tanaka, Akira; Takahashi, Haruhisa (2006), "The worst-case time complexity for generating all maximal cliques and computational experiments", Theoretical Computer Science, 363 (1): 28–42, doi:10.1016/j.tcs.2006.06.015
For the Brouwer–Haemers and Games graphs, see Bondarenko, Andriy V.; Radchenko, Danylo V. (2013), "On a family of strongly regular graphs with $\lambda =1$ ", Journal of Combinatorial Theory, Series B, 103 (4): 521–531, arXiv:1201.0383, doi:10.1016/j.jctb.2013.05.005, MR 3071380. For the Berlekamp–van Lint–Seidel and Games graphs, see van Lint, J. H.; Brouwer, A. E. (1984), "Strongly regular graphs and partial geometries" (PDF), in Jackson, David M.; Vanstone, Scott A. (eds.), Enumeration and Design: Papers from the conference on combinatorics held at the University of Waterloo, Waterloo, Ont., June 14–July 2, 1982, London: Academic Press, pp. 85–122, MR 0782310
Kalai, Gil (1989), "The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics, 5 (1): 389–391, doi:10.1007/BF01788696, MR 1554357, S2CID 8917264
von Koch, Helge (1904), "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire", Arkiv för Matematik (in French), 1: 681–704, JFM 35.0387.02
See, e.g., Mihăilă, Ioana (2004), "The rationals of the Cantor set", The College Mathematics Journal, 35 (4): 251–255, doi:10.2307/4146907, JSTOR 4146907, MR 2076132
Hinz, Andreas M.; Klavžar, Sandi; Milutinović, Uroš; Petr, Ciril (2013), "2.3 Hanoi graphs", The tower of Hanoi—myths and maths, Basel: Birkhäuser, pp. 120–134, doi:10.1007/978-3-0348-0237-6, ISBN 978-3-0348-0236-9, MR 3026271
Telser, L. G. (October 1995), "Optimal denominations for coins and currency", Economics Letters, 49 (4): 425–427, doi:10.1016/0165-1765(95)00691-8
Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L. (2003), "On perfect totient numbers", Journal of Integer Sequences, 6 (4), Article 03.4.5, Bibcode:2003JIntS...6...45I, MR 2051959
Sloane, N. J. A. (ed.), "Sequence A005836", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
Gupta, Hansraj (1978), "Powers of 2 and sums of distinct powers of 3", Univerzitet u Beogradu Publikacije Elektrotehničkog Fakulteta, Serija Matematika i Fizika (602–633): 151–158 (1979), MR 0580438
Gardner, Martin (November 1977), "In which joining sets of points leads into diverse (and diverting) paths", Scientific American, 237 (5): 18–28, Bibcode:1977SciAm.237e..18G, doi:10.1038/scientificamerican1177-18

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[1] Ranucci, Ernest R. (December 1968), "Tantalizing ternary", The Arithmetic Teacher, 15 (8): 718–722, doi:10.5951/AT.15.8.0718, JSTOR 41185884

[2] Moon, J. W.; Moser, L. (1965), "On cliques in graphs", Israel Journal of Mathematics, 3: 23–28, doi:10.1007/BF02760024, MR 0182577, S2CID 9855414

[3] Tomita, Etsuji; Tanaka, Akira; Takahashi, Haruhisa (2006), "The worst-case time complexity for generating all maximal cliques and computational experiments", Theoretical Computer Science, 363 (1): 28–42, doi:10.1016/j.tcs.2006.06.015

[4] For the Brouwer–Haemers and Games graphs, see Bondarenko, Andriy V.; Radchenko, Danylo V. (2013), "On a family of strongly regular graphs with $\lambda =1$ ", Journal of Combinatorial Theory, Series B, 103 (4): 521–531, arXiv:1201.0383, doi:10.1016/j.jctb.2013.05.005, MR 3071380. For the Berlekamp–van Lint–Seidel and Games graphs, see van Lint, J. H.; Brouwer, A. E. (1984), "Strongly regular graphs and partial geometries" (PDF), in Jackson, David M.; Vanstone, Scott A. (eds.), Enumeration and Design: Papers from the conference on combinatorics held at the University of Waterloo, Waterloo, Ont., June 14–July 2, 1982, London: Academic Press, pp. 85–122, MR 0782310

[5] Kalai, Gil (1989), "The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics, 5 (1): 389–391, doi:10.1007/BF01788696, MR 1554357, S2CID 8917264

[6] von Koch, Helge (1904), "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire", Arkiv för Matematik (in French), 1: 681–704, JFM 35.0387.02

[7] See, e.g., Mihăilă, Ioana (2004), "The rationals of the Cantor set", The College Mathematics Journal, 35 (4): 251–255, doi:10.2307/4146907, JSTOR 4146907, MR 2076132

[8] Hinz, Andreas M.; Klavžar, Sandi; Milutinović, Uroš; Petr, Ciril (2013), "2.3 Hanoi graphs", The tower of Hanoi—myths and maths, Basel: Birkhäuser, pp. 120–134, doi:10.1007/978-3-0348-0237-6, ISBN 978-3-0348-0236-9, MR 3026271

[9] Telser, L. G. (October 1995), "Optimal denominations for coins and currency", Economics Letters, 49 (4): 425–427, doi:10.1016/0165-1765(95)00691-8

[10] Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L. (2003), "On perfect totient numbers", Journal of Integer Sequences, 6 (4), Article 03.4.5, Bibcode:2003JIntS...6...45I, MR 2051959

[11] Sloane, N. J. A. (ed.), "Sequence A005836", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation

[12] Gupta, Hansraj (1978), "Powers of 2 and sums of distinct powers of 3", Univerzitet u Beogradu Publikacije Elektrotehničkog Fakulteta, Serija Matematika i Fizika (602–633): 151–158 (1979), MR 0580438

[13] Gardner, Martin (November 1977), "In which joining sets of points leads into diverse (and diverting) paths", Scientific American, 237 (5): 18–28, Bibcode:1977SciAm.237e..18G, doi:10.1038/scientificamerican1177-18