|
|
| Line 1: |
Line 1: |
| In [[mathematics]], a '''perfect power''' is a positive [[integer]] that can be expressed as an integer [[Exponentiation|power]] of another positive integer. More formally, ''n'' is a perfect power if there exist [[natural number]]s ''m'' > 1, and ''k'' > 1 such that ''m<sup>k''</sup> = ''n''. In this case, ''n'' may be called a '''perfect ''k''th power'''. If ''k'' = 2 or ''k'' = 3, then ''n'' is called a [[Square number|perfect square]] or [[perfect cube]], respectively. Sometimes 1 is also considered a perfect power (1<sup>''k''</sup> = 1 for any ''k'').
| | Let me at [http://www.onbizin.co.kr/xe/?document_srl=354357 home std test kit] home std test first start by [http://Bedsider.org/features/253 introducing] myself. My title is Boyd Butts although it is not the name on my birth certification. Managing individuals has been his day job for a while. It's not a common factor but what she likes performing is base jumping and now she is trying to earn cash [http://Browardchd.org/healths_sexually.aspx std testing] at home with it. For many years he's been living in North Dakota and his family enjoys it.<br><br>Take home std test kit a look at my weblog: [http://tfor.vectorsigma.ru/node/5409 at home std test] at home std testing ([http://www.bluecomtech.com/3dp/index.php?document_srl=701711&mid=bbs_firmware just click the next web page]) |
| | |
| == Examples and sums == | |
| A [[sequence]] of perfect powers can be generated by iterating through the possible values for ''m'' and ''k''. The first few ascending perfect powers in numerical order (showing duplicate powers) are {{OEIS|id=A072103}}:
| |
| :<math> 2^2 = 4,\ 2^3 = 8,\ 3^2 = 9,\ 2^4 = 16,\ 4^2 = 16,\ 5^2 = 25,\ 3^3 = 27,\ </math> <math> 2^5 = 32,\ 6^2 = 36,\ 7^2 = 49,\ 2^6 = 64,\ 4^3 = 64,\ 8^2 = 64, \dots </math>
| |
| | |
| The [[Series (mathematics)|sum]] of the [[Reciprocal (mathematics)|reciprocals]] of the perfect powers (including duplicates) is 1:
| |
| :<math>\sum_{m=2}^{\infty} \sum_{k=2}^{\infty}\frac{1}{m^k}=1.</math>
| |
| | |
| which can be proved as follows:
| |
| :<math>\sum_{m=2}^{\infty} \sum_{k=2}^{\infty}\frac{1}{m^k}
| |
| =\sum_{m=2}^{\infty} \frac {1}{m^2} \sum_{k=0}^{\infty}\frac{1}{m^k}
| |
| =\sum_{m=2}^{\infty} \frac {1}{m^2} \left( \frac{m}{m-1} \right)
| |
| =\sum_{m=2}^{\infty} \frac {1}{m(m-1)}
| |
| =\sum_{m=2}^{\infty} \left( \frac {1}{m-1} - \frac {1}{m} \right) = 1 \, .</math>
| |
| | |
| The first perfect powers without duplicates are ({{OEIS2C|id=A001597}}):
| |
| :(sometimes 1), 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, ...
| |
| | |
| The sum of the reciprocals of the perfect powers ''p'' without duplicates is:<ref>{{MathWorld|urlname=PerfectPower|title=Perfect Power}}</ref>
| |
| | |
| :<math>\sum_{p}\frac{1}{p}=\sum_{k=2}^{\infty}\mu(k)(1-\zeta(k)) \approx 0.874464368 \dots</math>
| |
| | |
| where μ(''k'') is the [[Möbius function]] and ζ(''k'') is the [[Riemann zeta function]].
| |
| | |
| According to [[Leonhard Euler|Euler]], [[Christian Goldbach|Goldbach]] showed (in a now lost letter) that the sum of 1/(''p''−1) over the set of perfect powers ''p'', excluding 1 and excluding duplicates, is 1:
| |
| | |
| :<math>\sum_{p}\frac{1}{p-1}= {\frac{1}{3} + \frac{1}{7} + \frac{1}{8}+ \frac{1}{15} + \frac{1}{24} + \frac{1}{26}+ \frac{1}{31}}+ \cdots = 1.</math>
| |
| | |
| This is sometimes known as the [[Goldbach-Euler theorem]].
| |
| | |
| The constant concatenating "0." with the base 10 representations of perfect powers in order will be irrational number.
| |
| | |
| ==Detecting perfect powers==
| |
| | |
| Detecting whether or not a given natural number ''n'' is a perfect power may be accomplished in many different ways, with varying levels of [[Computational complexity theory|complexity]]. One of the simplest such methods is to consider all possible values for ''k'' across each of the [[divisor]]s of ''n'', up to <math>k \leq \log_2 n</math>. So if the divisors of <math>n</math> are <math>n_1, n_2, \dots, n_j</math> then one of the values <math>n_1^2, n_2^2, \dots, n_j^2, n_1^3, n_2^3, \dots</math> must be equal to ''n'' if ''n'' is indeed a perfect power.
| |
| | |
| This method can immediately be simplified by instead considering only [[Prime number|prime]] values of ''k''. This is because if <math>n = m^k</math> for a [[composite number|composite]] <math>k = ap</math> where ''p'' is prime, then this can simply be rewritten as <math>n = m^k = m^{ap} = (m^a)^p</math>. Because of this result, the [[Minimal element|minimal]] value of ''k'' must necessarily be prime.
| |
| | |
| If the full factorization of ''n'' is known, say <math>n = p_1^{\alpha_1}p_2^{\alpha_2} \cdots p_r^{\alpha_r}</math> where the <math>p_i</math> are distinct primes, then ''n'' is a perfect power [[if and only if]] <math>\gcd(\alpha_1, \alpha_2, \ldots, \alpha_r) > 1</math> where gcd denotes the [[greatest common divisor]]. As an example, consider ''n'' = 2<sup>96</sup>·3<sup>60</sup>·7<sup>24</sup>. Since gcd(96, 60, 24) = 12, ''n'' is a perfect 12th power (and a perfect 6th power, 4th power, cube and square, since 6, 4, 3 and 2 divide 12).
| |
| | |
| ==Gaps between perfect powers==
| |
| In 2002 Romanian mathematician [[Preda Mihăilescu]] proved that the only pair of consecutive perfect powers is 2<sup>3</sup> = 8 and 3<sup>2</sup> = 9, thus proving [[Catalan's conjecture]].
| |
| | |
| Pillai's conjecture states that for any given positive integer ''k'' there are only a finite number of pairs of perfect powers whose difference is ''k''. This is an unsolved problem.<ref>{{MathWorld|urlname=PillaisConjecture|title=Pillai's Conjecture}}</ref>
| |
| | |
| ==Calculation by Recursion for positive integers==
| |
| This being an alternate way to calculate perfect powers has yet to be found useful. It is based on the observation that the difference between a<sup>b</sup> and (a+1)<sup>b</sup> where a > b may not be constant, but if you take the difference of successive differences, ''b'' times, there is a constant ''b!'' factor. For example, 9<sup>4</sup> = 6561, and 10<sup>4</sup> is 10000. the difference is 3439. The difference between 8<sup>4</sup> and 9<sup>4</sup> is 2465, meaning the difference of differences is 974. A step further and you have 204. One step further, and you have 24, which is equal to 4!. One step further and collating this 'key' row from progressively larger exponents yields a triangle similar to Pascal's, but with a differing formula for generation. A part of this table is shown below:
| |
| | |
| Define the following function on the range of positive integers:
| |
| :<math>K(a,b) = 1</math> where ''a'' = 1 or ''a'' = ''b''
| |
| :<math>K(a,b) = 0</math> where ''b > a''
| |
| :<math>K(a,b) = bK(a-b,b) + (a-b+1)K(a-b+1,b-1)</math> elsewhere
| |
| | |
| This function generates the following output:
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| | || 1 || 2 || 3 || 4 || 5 || 6
| |
| |-
| |
| | 1 || 1 || 0|| 0|| 0|| 0|| 0
| |
| |-
| |
| | 2 || 1|| 1|| 0|| 0|| 0|| 0
| |
| |-
| |
| | 3 || 1|| 4|| 1|| 0|| 0|| 0
| |
| |-
| |
| | 4 || 1|| 11|| 11|| 1|| 0|| 0
| |
| |-
| |
| | 5 || 1|| 26|| 66|| 26|| 1|| 0
| |
| |-
| |
| | 6 || 1|| 57|| 302|| 302|| 57|| 1
| |
| |}
| |
| | |
| Also define the following function on the range of positive integers:
| |
| (This is very closely related to the Binomial Theorem and Pascal's Triangle)
| |
| :<math>P(a,b) = 1</math> where ''a'' = 1 or ''b'' = 1
| |
| :<math>P(a,b) = P(a-1, b) + P(a, b-1)</math> elsewhere
| |
| | |
| The table this generates can be seen as pascal's triangle fallen over to the left, so that what were rows on Pascal's triangle have become diagonal series in the table.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| | || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8
| |
| |-
| |
| | 1|| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1
| |
| |-
| |
| | 2|| 1 || 2|| 3|| 4|| 5|| 6|| 7|| 8
| |
| |-
| |
| | 3|| 1 || 3|| 6|| 10 || 15 || 21|| 28|| 36
| |
| |-
| |
| | 4|| 1 || 4|| 10 || 20|| 35|| 56|| 84|| 120
| |
| |-
| |
| | 5|| 1 || 5|| 15|| 35|| 70|| 126 || 210|| 330
| |
| |-
| |
| | 6|| 1 || 6|| 21|| 56|| 126|| 252|| 462|| 792
| |
| |-
| |
| | 7|| 1 || 7|| 28|| 84|| 210|| 462|| 924|| 1716
| |
| |-
| |
| | 8|| 1 || 8|| 36|| 120|| 330|| 792|| 1716|| 3432
| |
| | |
| |}
| |
| | |
| It can then be stated that:
| |
| :<math>a^b = \sum_{x=1}^b \,\! P ( a - x + 1 , b + 1 ) K ( b , x )</math>
| |
| | |
| Example:
| |
| :<math>7^3 = \sum_{x=1}^b P(8 - x, 4)K(3,x) = P(7, 4)K(3,1) + P(6,4)K(3,2) + P(5,4)K(3,3)</math>
| |
| Expanding P(7,4)
| |
| :<math>P(7,4) = P(7,3) + P(6,4) P(7,2) + 2P(6,3) + P(5,4)</math>
| |
| :<math>= P(7,1) + 3P(6,2) + 3P(5,3) + P(4,4)</math>
| |
| :<math>= 1 + 3P(6,1) + 6P(5,2) + 4P(4,3) + P(3,4) </math>
| |
| :<math>= 4 + 6P(5,1) + 10P(4,2) + 5P(3,3) + P(2,4) </math>
| |
| :<math>= 10 + 10P(4,1) + 15P(3,2) + 6P(2,3) + P(1,4) </math>
| |
| :<math>= 21 + 15P(3,1) + 21P(2,2) + 6P(1,3) </math>
| |
| :<math>= 42 + 21P(2,1) + 21P(1,2) = 84</math>
| |
| | |
| Or you can look up the values on the table and get P(6,4) = 56, and P(5,4) = 35.
| |
| | |
| By definition, K(3,1) = 1. Expanding K(3,2)
| |
| :<math>K(3,2) = 2K(1, 2) + 2K(2,1) = 4</math> | |
| | |
| By definition, K(3,3) = 1.
| |
| | |
| :<math>7^3 = P(7,4)K(3,1) + P(6,4)K(3,2) + P(5,4)K(3,3) = 84*1 + 56 * 4 + 35 * 1 = 343</math>
| |
| :<math>7^4 = P(7,5)K(4,1) + P(6,5)K(4,2) + P(5,5)K(4,3) + P(4,5)K(4,4) = 210*1+126*11+70*11+35*1 = 2401</math>
| |
| :<math>7^5 = 462*1+252*26+126*66+56*26+21*1 = 16807</math>
| |
| | |
| This calculation method can be used for all integer power calculations, as negative integers act the same way, simply applying he negative if the exponent is odd.
| |
| | |
| ==See also==
| |
| * [[Prime power]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| * {{cite journal | author=Daniel J. Bernstein | title=Detecting perfect powers in essentially linear time | journal=Mathematics of Computation | year=1998 | volume=67 | issue=223 | pages= 1253–1283 | url=http://cr.yp.to/papers/powers-ams.pdf | doi=10.1090/S0025-5718-98-00952-1 }}
| |
| | |
| == External links ==
| |
| *[http://www.recercat.net/bitstream/2072/920/1/776.pdf On a series of Goldbach and Euler]
| |
| | |
| | |
| {{Divisor classes}}
| |
| {{Classes of natural numbers}}
| |
| | |
| [[Category:Number theory]]
| |
| [[Category:Integer sequences]]
| |