mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a power of three is a number of the form where is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

, that is, the result of

exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...

with number

three 3 (three) is a number, numeral and digit. It is the natural number following 2 and preceding 4, and is the smallest odd prime number and the only prime preceding a square number. It has religious and cultural significance in many societies ...

as the base and integer as the

exponent In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...

. The first seven non-negative powers of three are: 1, 3, 9, 27, 81,

243 __NOTOC__ Year 243 ( CCXLIII) was a common year starting on Sunday of the Julian calendar. At the time, it was known in Rome as the Year of the Consulship of Arrianus and Papus (or, less frequently, year 996 ''Ab urbe condita''). The denominat ...

729 Year 729 ( DCCXXIX) was a common year starting on Saturday of the Julian calendar, the 729th year of the Common Era (CE) and Anno Domini (AD) designations, the 729th year of the 1st millennium, the 29th year of the 8th century, and the 10th and ...

, etc. (sequence A000244 in

OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...

)

Applications

The powers of three give the place values in the

ternary numeral system A ternary numeral system (also called base 3 or trinary) has 3 (number), three as its radix, base. Analogous to a bit, a ternary numerical digit, digit is a trit (trinary digit). One trit is equivalent to binary logarithm, log2 3 (about 1.5 ...

Graph theory

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, powers of three appear in the Moon–Moser bound on the number of

maximal independent set In graph theory, a maximal independent set (MIS) or maximal stable set is an Independent set (graph theory), independent set that is not a subset of any other independent set. In other words, there is no Vertex (graph theory), vertex outside th ...

s of an -vertex

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

, and in the time analysis of the

Bron–Kerbosch algorithm In computer science, the Bron–Kerbosch algorithm is an enumeration algorithm for finding all maximal cliques in an undirected graph. That is, it lists all subsets of vertices with the two properties that each pair of vertices in one of the liste ...

for finding these sets. Several important

strongly regular graph In graph theory, a strongly regular graph (SRG) is a regular graph with vertices and degree such that for some given integers \lambda, \mu \ge 0 * every two adjacent vertices have common neighbours, and * every two non-adjacent vertices h ...

s also have a number of vertices that is a power of three, including the

Brouwer–Haemers graph In the mathematical field of graph theory, the Brouwer–Haemers graph is a 20- regular undirected graph with 81 vertices and 810 edges. It is a strongly regular graph, a distance-transitive graph, and a Ramanujan graph. Although its construc ...

(81 vertices), Berlekamp–van Lint–Seidel graph (243 vertices), and

Games graph In graph theory, the Games graph is the largest known locally linear strongly regular graph. Its parameters as a strongly regular graph are (729,112,1,20). This means that it has 729 vertices, and 40824 edges (112 per vertex). Each edge is in a u ...

(729 vertices).

Enumerative combinatorics

enumerative combinatorics Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an inf ...

, there are signed subsets of a set of elements. In

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral co ...

, the

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

and all other

Hanner polytope In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Swedish mathematician Olof Hanner, who introduced them in 1956.. Construction The Hann ...

s have a number of faces (not counting the

empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...

as a face) that is a power of three. For example, a , or

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

, has 4 vertices, 4 edges and 1 face, and . Kalai's conjecture states that this is the minimum possible number of faces for a

centrally symmetric In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or ...

polytope.

Inverse power of three lengths

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

and

fractal geometry In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ...

, inverse power-of-three lengths occur in the constructions leading to the

Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Cur ...

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...

, Sierpinski carpet and

Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sie ...

, in the number of elements in the construction steps for a Sierpinski triangle, and in many formulas related to these sets. There are possible states in an -disk Tower of Hanoi puzzle or vertices in its associated Hanoi graph. In a balance puzzle with weighing steps, there are 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

coin A coin is a small object, usually round and flat, used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to facilitate trade. They are most often issued by ...

Perfect totient numbers

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, all powers of three are perfect totient numbers. The sums of distinct powers of three form a

Stanley sequence In mathematics, a Stanley sequence is an integer sequence generated by a greedy algorithm that chooses the sequence members to avoid arithmetic progressions. If S is a finite set of non-negative integers on which no three elements form an arithmeti ...

, the lexicographically smallest sequence that does not contain an

arithmetic progression An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...

of three elements. A

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

Paul Erdős Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...

states that this sequence contains no

powers of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^n(1). In the Hardy hi ...

other than 1, 4, and 256.

Graham's number

Graham's number Graham's number is an Large numbers, immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes's number and Moser's number, bot ...

, an enormous number arising from a

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in R ...

, is (in the version popularized by

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writin ...

) a power of three. However, the actual publication of the proof by

Ronald Graham Ronald Lewis Graham (October 31, 1935July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He ...

used a different number which is a

power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number 2, two as the Base (exponentiation), base and integer as the exponent. In the fast-growing hierarchy, is exactly equal to f_1^ ...

and much smaller.

References

{{DEFAULTSORT:Power Of Three Integers 3 (number)