Inmathematics, apower of three is a number of the form3ⁿ wheren is aninteger, that is, the result ofexponentiation with numberthree as thebase and integer n as theexponent. The first ten non-negative powers of three are:

1,3,9,27,81,243,729, 2187, 6561, 19683, etc. (sequenceA000244 inOEIS)

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

Ingraph theory, powers of three appear in the Moon–Moser bound3^n/3 on the number ofmaximal independent sets of ann-vertexgraph,^[2] and in the time analysis of theBron–Kerbosch algorithm for finding these sets.^[3] Several importantstrongly regular graphs also have a number of vertices that is a power of three, including theBrouwer–Haemers graph (81 vertices),Berlekamp–van Lint–Seidel graph (243 vertices), andGames graph (729 vertices).^[4]

Inenumerative combinatorics, there are3ⁿsigned subsets of a set ofn elements. Inpolyhedral combinatorics, thehypercube and all otherHanner polytopes have a number of faces (not counting theempty set as a face) that is a power of three. For example, a2-cube, orsquare, has 4 vertices, 4 edges and 1 face, and4 + 4 + 1 = 3².Kalai's3^d conjecture states that this is the minimum possible number of faces for acentrally symmetric polytope.^[5]

Inrecreational mathematics andfractal geometry, inverse power-of-three lengths occur in the constructions leading to theKoch snowflake,^[6]Cantor set,^[7]Sierpinski carpet andMenger sponge, in the number of elements in the construction steps for aSierpinski triangle, and in many formulas related to these sets. There are3ⁿ possible states in ann-diskTower of Hanoi puzzle or vertices in its associatedHanoi graph.^[8] In abalance puzzle withw weighing steps, there are3^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 ofcoins.^[9]

Innumber theory, all powers of three areperfect totient numbers.^[10] The sums of distinct powers of three form aStanley sequence, the lexicographically smallest sequence that does not contain anarithmetic progression of three elements.^[11] Aconjecture ofPaul Erdős states that this sequence contains nopowers of two other than 1, 4, and 256.^[12]

Graham's number, an enormous number arising from aproof inRamsey theory, is (in the version popularized byMartin Gardner) a power of three.However, the actual publication of the proof byRonald Graham used a different number which is apower of two and much smaller.^[13]

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

• Integers
• 3 (number)

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