Power of three
In mathematics, a power of three is a number of the form $3^n, where n is an integer, obtained through exponentiation with base 3 and exponent n.[1] This operation represents repeated multiplication of 3 by itself n times when n is positive, yielding 1 when n=0, and reciprocals $1/3^{|n|} when n is negative.[2] The sequence of powers of three, often considered for non-negative integers n, begins as 1, 3, 9, 27, 81, 243, 729, 2187, and continues indefinitely, with each subsequent term obtained by multiplying the previous by 3.[3] These numbers exhibit properties such as being odd for all n \geq 1 and serving as perfect totient numbers, meaning the sum from k=1 to $3^n of Euler's totient function \phi(k) equals $3^n.[3] In base-10 representation, powers of three grow exponentially, with $3^{10} = 59049 and $3^{20} \approx 3.48 \times 10^9, illustrating their rapid increase. Powers of three play key roles in several areas of mathematics and computing. In number theory, every integer has a unique representation in balanced ternary, a numeral system using digits -1, 0, and 1 as coefficients of powers of three, enabling efficient arithmetic without a separate sign bit.[4] This system contrasts with standard ternary (base 3 with digits 0-2) and has historical applications in early computing designs, such as ternary logic circuits.[4] Additionally, powers of three appear in graph theory contexts, such as counting certain tree structures or in algorithms for sequence analysis, and in combinatorics for partitioning problems involving multiples of 3.[3]Definition and Fundamentals
Definition
In mathematics, a power of three is defined as a number of the form $3^n, where n is an integer and the base 3 is raised to the exponent n through exponentiation.[5] Exponentiation in this context represents the process of multiplying the base by itself n times when n is positive; for instance, $3^3 = 3 \times 3 \times 3 = 27.[6] This sequence, known as the powers of 3, is cataloged in the Online Encyclopedia of Integer Sequences (OEIS) as A000244.[3] The first ten terms of the sequence are:$3^0 = 1,
$3^1 = [3](/page/3),
$3^2 = 9,
$3^3 = [27](/page/27),
$3^4 = 81,
$3^5 = 243,
$3^6 = [729](/page/729),
$3^7 = [2187](/page/21-87),
$3^8 = 6561,
$3^9 = 19683.[3] For completeness, powers of three extend to negative exponents, where $3^{-n} = \frac{1}{3^n} for positive integer n, resulting in fractions like $3^{-1} = \frac{1}{[3](/page/3)} and $3^{-2} = \frac{1}{9}.[5] However, the concept primarily emphasizes non-negative exponents in integer contexts. Powers of three form the basis for place values in the ternary numeral system.[7]
Notation and Sequence
In mathematics, powers of three are denoted using exponential notation as $3^n, where n is an integer and the exponent appears as a superscript to the right of the base 3.[8] This compact form represents the product of n factors of 3 when n is positive, with $3^0 = 1 by convention.[9] In computational and programming contexts, such as in C++ or Python, the operation is commonly expressed using the power function pow(3, n).[10] The infinite sequence of powers of three, given by a_n = 3^n for n = 0, 1, 2, \dots, is cataloged as A000244 in the On-Line Encyclopedia of Integer Sequences (OEIS).[3] This sequence is strictly increasing for n \geq 0, as each term is precisely three times the preceding one, resulting in exponential growth.[3] Relative to the sequence of powers of two ($2^n), powers of three grow more rapidly owing to the larger base, yielding values that surpass those of $2^n for sufficiently large n.[11] Higher powers of three can be computed through iterative multiplication, initializing with $3^0 = 1 and successively multiplying by 3 for each increment in the exponent: $3^k = 3 \times 3^{k-1}.[12] For approximating large exponents without exact computation, the properties of logarithms provide a useful tool; specifically, \log_{10}(3^n) = n \log_{10} 3 \approx n \times 0.4771, which estimates the number of digits and magnitude as $3^n \approx 10^{n \times 0.4771}.[8] The table below lists exact values of $3^n for n from 0 to 20, beyond which scientific notation is practical for representation.[3]| n | $3^n |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 9 |
| 3 | 27 |
| 4 | 81 |
| 5 | 243 |
| 6 | 729 |
| 7 | 2,187 |
| 8 | 6,561 |
| 9 | 19,683 |
| 10 | 59,049 |
| 11 | 177,147 |
| 12 | 531,441 |
| 13 | 1,594,323 |
| 14 | 4,782,969 |
| 15 | 14,348,907 |
| 16 | 43,046,721 |
| 17 | 129,140,163 |
| 18 | 387,420,489 |
| 19 | 1,162,261,467 |
| 20 | 3,486,784,401 |