Power of three

In mathematics, a power of three is a number of the form Template:Math where Template:Mvar is an integer, that is, the result of exponentiation with number three as the base and integer Template:Mvar as the exponent.

In base 10, every power of 3 has an even number as its second-last digit.

Applications

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 Template:Math on the number of maximal independent sets of an Template:Mvar-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 Template:Math signed subsets of a set of Template:Mvar 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 Template:Nowrap, or square, has 4 vertices, 4 edges and 1 face, and Template:Math. [[Kalai's 3^d conjecture|Kalai's Template:Math 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 Template:Math possible states in an Template:Mvar-disk Tower of Hanoi puzzle or vertices in its associated Hanoi graph.^[8] In a balance puzzle with Template:Mvar weighing steps, there are Template:Math 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]