power of three

The powers of three give the place values in the ternary numeral system.{{citation

| last = Ranucci | first = Ernest R.

| date = December 1968

| issue = 8

| journal = The Arithmetic Teacher

| jstor = 41185884

| pages = 718–722

| title = Tantalizing ternary

| volume = 15| doi = 10.5951/AT.15.8.0718

}}

In graph theory, powers of three appear in the Moon–Moser bound {{math|3^n/3}} on the number of maximal independent sets of an {{mvar|n}}-vertex graph,{{citation | last1 = Moon | first1 = J. W. | author2-link = Leo Moser | last2 = Moser | first2 = L. | title = On cliques in graphs | journal = Israel Journal of Mathematics | volume = 3 | year = 1965 | pages = 23–28 |mr=0182577 | doi = 10.1007/BF02760024 | doi-access= | s2cid = 9855414 }} and in the time analysis of the Bron–Kerbosch algorithm for finding these sets.{{citation |first1=Etsuji |last1=Tomita |first2=Akira |last2=Tanaka |first3=Haruhisa |last3=Takahashi |title=The worst-case time complexity for generating all maximal cliques and computational experiments |journal=Theoretical Computer Science |volume=363 |issue=1 |pages=28–42 |year=2006 |doi=10.1016/j.tcs.2006.06.015|doi-access= }} 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).For the Brouwer–Haemers and Games graphs, see {{citation

| last1 = Bondarenko | first1 = Andriy V.

| last2 = Radchenko | first2 = Danylo V.

| arxiv = 1201.0383

| doi = 10.1016/j.jctb.2013.05.005 | doi-access=free

| issue = 4

| journal = Journal of Combinatorial Theory

| mr = 3071380

| pages = 521–531

| series = Series B

| title = On a family of strongly regular graphs with $\backslash lambda=1$

| volume = 103

| year = 2013}}. For the Berlekamp–van Lint–Seidel and Games graphs, see {{citation

| last1 = van Lint | first1 = J. H. | author1-link = J. H. van Lint

| last2 = Brouwer | first2 = A. E. | author2-link = Andries Brouwer

| editor1-last = Jackson | editor1-first = David M. | editor1-link = David M. Jackson

| editor2-last = Vanstone | editor2-first = Scott A. | editor2-link = Scott Vanstone

| contribution = Strongly regular graphs and partial geometries

| contribution-url = https://pure.tue.nl/ws/files/2394798/595248.pdf

| location = London

| mr = 782310

| pages = 85–122

| publisher = Academic Press

| title = Enumeration and Design: Papers from the conference on combinatorics held at the University of Waterloo, Waterloo, Ont., June 14–July 2, 1982

| year = 1984}}

In enumerative combinatorics, there are {{math|3ⁿ}} signed subsets of a set of {{mvar|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 {{nowrap|2-cube}}, or square, has 4 vertices, 4 edges and 1 face, and {{math|1=4 + 4 + 1 = 3²}}. Kalai's 3^d conjecture states that this is the minimum possible number of faces for a centrally symmetric polytope.{{citation

| last = Kalai | first = Gil | author-link = Gil Kalai

| doi = 10.1007/BF01788696

| issue = 1

| journal = Graphs and Combinatorics

| mr = 1554357

| pages = 389–391

| title = The number of faces of centrally-symmetric polytopes

| volume = 5

| year = 1989| s2cid = 8917264 }}

In recreational mathematics and fractal geometry, inverse power-of-three lengths occur in the constructions leading to the Koch snowflake,{{citation

| last = von Koch | first = Helge | author-link = Helge von Koch

| jfm = 35.0387.02

| journal = Arkiv för Matematik

| language = fr

| pages = 681–704

| title = Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire

| url = https://babel.hathitrust.org/cgi/pt?id=inu.30000100114564;view=1up;seq=673

| volume = 1

| year = 1904}} Cantor set,See, e.g., {{citation

| last = Mihăilă | first = Ioana

| doi = 10.2307/4146907

| issue = 4

| journal = The College Mathematics Journal

| mr = 2076132

| pages = 251–255

| title = The rationals of the Cantor set

| volume = 35

| year = 2004| jstor = 4146907

}} 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 {{math|3ⁿ}} possible states in an {{mvar|n}}-disk Tower of Hanoi puzzle or vertices in its associated Hanoi graph.{{citation

| last1 = Hinz | first1 = Andreas M.

| last2 = Klavžar | first2 = Sandi | author2-link = Sandi Klavžar

| last3 = Milutinović | first3 = Uroš

| last4 = Petr | first4 = Ciril

| contribution = 2.3 Hanoi graphs

| doi = 10.1007/978-3-0348-0237-6

| isbn = 978-3-0348-0236-9

| mr = 3026271

| pages = 120–134

| publisher = Birkhäuser | location = Basel

| title = The tower of Hanoi—myths and maths

| title-link = The Tower of Hanoi – Myths and Maths

| year = 2013}} In a balance puzzle with {{mvar|w}} weighing steps, there are {{math|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.{{citation

| last = Telser | first = L. G. | author-link = Lester G. Telser

| date = October 1995

| doi = 10.1016/0165-1765(95)00691-8

| issue = 4

| journal = Economics Letters

| pages = 425–427

| title = Optimal denominations for coins and currency

| volume = 49}}

In number theory, all powers of three are perfect totient numbers.{{citation

| last1 = Iannucci | first1 = Douglas E.

| last2 = Deng | first2 = Moujie

| last3 = Cohen | first3 = Graeme L.

| at = Article 03.4.5

| issue = 4

| journal = Journal of Integer Sequences

| mr = 2051959

| title = On perfect totient numbers

| url = https://cs.uwaterloo.ca/journals/JIS/VOL6/Cohen2/cohen50.html

| volume = 6

| year = 2003| bibcode = 2003JIntS...6...45I

}} 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.{{cite OEIS|A005836|mode=cs2}} A conjecture of Paul Erdős states that this sequence contains no powers of two other than 1, 4, and 256.{{citation

| last = Gupta | first = Hansraj

| issue = 602–633

| journal = Univerzitet u Beogradu Publikacije Elektrotehničkog Fakulteta, Serija Matematika i Fizika

| mr = 580438

| pages = 151–158 (1979)

| title = Powers of 2 and sums of distinct powers of 3

| year = 1978}}

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.{{citation

| last = Gardner | first = Martin | author-link = Martin Gardner

| date = November 1977

| issue = 5

| journal = Scientific American

| pages = 18–28

| title = In which joining sets of points leads into diverse (and diverting) paths

| volume = 237| doi = 10.1038/scientificamerican1177-18 | bibcode = 1977SciAm.237e..18G }}

• Power of 10
• Power of two
• Square root of 3

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{{Series (mathematics)}}

{{Classes of natural numbers}}

{{Large numbers}}

{{DEFAULTSORT:Power Of Three}}

Category:Integers

Category:3 (number)