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| A '''powerful number''' is a [[positive integer]] ''m'' such that for every [[prime number]] ''p'' dividing ''m'', ''p''<sup>2</sup> also divides ''m''. Equivalently, a powerful number is the product of a [[Square number|square]] and a [[Cube (arithmetic)|cube]], that is, a number ''m'' of the form ''m'' = ''a''<sup>2</sup>''b''<sup>3</sup>, where ''a'' and ''b'' are positive integers. Powerful numbers are also known as '''squareful''', '''square-full''', or '''2-full'''. [[Paul Erdős]] and [[George Szekeres]] studied such numbers and [[Solomon W. Golomb]] named such numbers ''powerful''.
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| The following is a list of all powerful numbers between 1 and 1000:
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| :1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000 {{OEIS|id=A001694}}.
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| == Equivalence of the two definitions ==
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| If ''m'' = ''a''<sup>2</sup>''b''<sup>3</sup>, then every prime in the [[prime factorization]] of ''a'' appears in the prime factorization of ''m'' with an exponent of at least two, and every prime in the prime factorization of ''b'' appears in the prime factorization of ''m'' with an exponent of at least three; therefore, ''m'' is powerful.
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| In the other direction, suppose that ''m'' is powerful, with prime factorization
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| :<math>m = \prod p_i^{\alpha_i},</math>
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| where each α<sub>i</sub> ≥ 2. Define γ<sub>i</sub> to be three if α<sub>i</sub> is odd, and zero otherwise, and define β<sub>i</sub> = α<sub>i</sub> - γ<sub>i</sub>. Then, all values β<sub>i</sub> are nonnegative even integers, and all values γ<sub>i</sub> are either zero or three, so
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| :<math>m = (\prod p_i^{\beta_i})(\prod p_i^{\gamma_i}) = (\prod p_i^{\beta_i/2})^2(\prod p_i^{\gamma_i/3})^3</math>
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| supplies the desired representation of ''m'' as a product of a square and a cube.
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| Informally, given the prime factorization of ''m'', take ''b'' to be the product of the prime factors of ''m'' that have an odd exponent (if there are none, then take ''b'' to be 1). Because ''m'' is powerful, each prime factor with an odd exponent has an exponent that is at least 3, so ''m''/''b''<sup>3</sup> is an integer. In addition, each prime factor of ''m''/''b''<sup>3</sup> has an even exponent, so ''m''/''b''<sup>3</sup> is a perfect square, so call this ''a''<sup>2</sup>; then ''m'' = ''a''<sup>2</sup>''b''<sup>3</sup>. For example:
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| <br>
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| :<math>m = 21600 = 2^5 \times 3^3 \times 5^2 \, ,</math>
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| :<math>b = 2 \times 3 = 6 \, ,</math>
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| :<math>a = \sqrt{\frac{m}{b^3}} = \sqrt{2^2 \times 5^2} = 10 \, ,</math>
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| :<math>m = a^2b^3 = 10^2 \times 6^3 \, .</math>
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| The representation ''m'' = ''a''<sup>2</sup>''b''<sup>3</sup> calculated in this way has the property that ''b'' is [[Square-free integer|squarefree]], and is uniquely defined by this property. | |
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| == Mathematical properties ==
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| The [[Dirichlet series]] generating function for powerful numbers is
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| :<math>\frac{\zeta(2s)\zeta(3s)}{\zeta(6s)} \ </math>
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| and so the sum of reciprocals of powerful numbers converges to
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| :<math>\prod_p\left(1+\frac{1}{p(p-1)}\right)=\frac{\zeta(2)\zeta(3)}{\zeta(6)} = \frac{315}{2\pi^4}\zeta(3),</math>
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| where ''p'' runs over all primes, ζ(''s'') denotes the [[Riemann zeta function]], and ζ(3) is [[Apéry's constant]] (Golomb, 1970).
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| Let ''k''(''x'') denote the number of powerful numbers in the interval [1,''x'']. Then ''k''(''x'') is proportional to the [[square root]] of ''x''. More precisely,
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| :<math>cx^{1/2}-3x^{1/3}\le k(x) \le cx^{1/2}, c=\zeta(3/2)/\zeta(3)=2.173\cdots</math> | |
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| (Golomb, 1970).
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| The two smallest consecutive powerful numbers are 8 and 9. Since [[Pell's equation]] ''x''<sup>2</sup> − 8''y''<sup>2</sup> = 1 has infinitely many integral solutions, there are infinitely many pairs of consecutive powerful numbers (Golomb, 1970); more generally, one can find consecutive powerful numbers by solving a similar Pell equation ''x''<sup>2</sup> − ''ny''<sup>2</sup> = ±1 for any [[perfect cube]] ''n''. However, one of the two powerful numbers in a pair formed in this way must be a square. According to Guy, Erdős has asked whether there are infinitely many pairs of consecutive powerful numbers such as (23<sup>3</sup>, 2<sup>3</sup>3<sup>2</sup>13<sup>2</sup>) in which neither number in the pair is a square. Jaroslaw Wroblewski showed that there are indeed infinitely many such pairs by showing that 3<sup>3</sup>c<sup>2</sup>+1=7<sup>3</sup>d<sup>2</sup> has infinitely many solutions. It is a [[Erdős conjecture|conjecture]] of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers.
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| == Sums and differences of powerful numbers ==
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| Any odd number is a difference of two consecutive squares: (''k'' + 1)<sup>2</sup> = ''k''<sup>2</sup> + 2k +1<sup>2</sup>, so (''k'' + 1)<sup>2</sup> - ''k''<sup>2</sup> = 2''k'' + 1. Similarly, any multiple of four is a difference of the squares of two numbers that differ by two: (''k'' + 2)<sup>2</sup> - ''k''<sup>2</sup> = 4''k'' + 4. However, a [[singly even number]], that is, a number divisible by two but not by four, cannot be expressed as a difference of squares. This motivates the question of determining which singly even numbers can be expressed as differences of powerful numbers. Golomb exhibited some representations of this type:
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| :2 = 3<sup>3</sup> − 5<sup>2</sup>
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| :10 = 13<sup>3</sup> − 3<sup>7</sup>
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| :18 = 19<sup>2</sup> − 7<sup>3</sup> = 3<sup>2</sup>(3<sup>3</sup> − 5<sup>2</sup>).
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| It had been conjectured that 6 cannot be so represented, and Golomb conjectured that there are infinitely many integers which cannot be represented as a difference between two powerful numbers. However, Narkiewicz showed that 6 can be so represented in infinitely many ways such as
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| :6 = 5<sup>4</sup>7<sup>3</sup> − 463<sup>2</sup>,
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| and McDaniel showed that every integer has infinitely many such representations (McDaniel, 1982).
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| [[Paul Erdős|Erdős]] conjectured that every sufficiently large integer is a sum of at most three powerful numbers; this was proved by [[Roger Heath-Brown]] (1987).
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| == Generalization == | |
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| More generally, we can consider the integers all of whose prime factors have exponents at least ''k''. Such an integer is called a ''k''-powerful number, ''k''-ful number, or ''k''-full number.
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| :(2<sup>''k''+1</sup> − 1)<sup>''k''</sup>, 2<sup>''k''</sup>(2<sup>''k''+1</sup> − 1)<sup>''k''</sup>, (2<sup>''k''+1</sup> − 1)<sup>''k''+1</sup>
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| are ''k''-powerful numbers in an [[arithmetic progression]]. Moreover, if ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''s''</sub> are ''k''-powerful in an arithmetic progression with common difference ''d'', then
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| : ''a''<sub>1</sub>(''a''<sub>''s''</sub> + ''d'')<sup>''k''</sup>, | |
| ''a''<sub>2</sub>(''a''<sub>''s''</sub> + ''d'')<sup>''k''</sup>, ..., ''a''<sub>''s''</sub>(''a''<sub>''s''</sub> + ''d'')<sup>''k''</sup>, (''a''<sub>''s''</sub> + d)<sup>''k''+1</sup>
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| are ''s'' + 1 ''k''-powerful numbers in an arithmetic progression.
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| We have an identity involving ''k''-powerful numbers:
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| :''a''<sup>''k''</sup>(''a''<sup>''l''</sup> + ... + 1)<sup>''k''</sup> + ''a''<sup>''k'' + 1</sup>(''a''<sup>''l''</sup> + ... + 1)<sup>''k''</sup> + ... + ''a''<sup>''k'' + ''l''</sup>(''a''<sup>''l''</sup> + ... + 1)<sup>''k''</sup> = ''a''<sup>''k''</sup>(''a''<sup>''l''</sup> + ... +1)<sup>''k''+1</sup>.
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| This gives infinitely many ''l''+1-tuples of ''k''-powerful numbers whose sum is also ''k''-powerful. Nitaj shows there are infinitely many solutions of ''x''+''y''=''z'' in relatively prime 3-powerful numbers(Nitaj, 1995). Cohn constructs an infinite family of solutions of ''x''+''y''=''z'' in relatively prime non-cube 3-powerful numbers as follows: the triplet
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| :''X'' = 9712247684771506604963490444281, ''Y'' = 32295800804958334401937923416351, ''Z'' = 27474621855216870941749052236511
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| is a solution of the equation 32''X''<sup>3</sup> + 49''Y''<sup>3</sup> = 81''Z''<sup>3</sup>. We can construct another solution by setting ''X''′ = ''X''(49''Y''<sup>3</sup> + 81''Z''<sup>3</sup>), ''Y''′ = −''Y''(32''X''<sup>3</sup> + 81''Z''<sup>3</sup>), ''Z''′ = ''Z''(32''X''<sup>3</sup> − 49''Y''<sup>3</sup>) and omitting the common divisor.
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| == See also ==
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| *[[Achilles number]]
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| == References ==
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| {{reflist}}
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| * {{cite journal
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| | author = Cohn, J. H. E.
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| | title = A conjecture of Erdős on 3-powerful numbers
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| | journal = Math. Comp.
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| | volume = 67
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| | year = 1998
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| | pages = 439–440
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| | url = http://www.ams.org/mcom/1998-67-221/S0025-5718-98-00881-3/ | doi = 10.1090/S0025-5718-98-00881-3
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| | issue = 221
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| }}
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| * {{cite journal
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| | author = [[Paul Erdős|Erdős, Paul]] and [[George Szekeres|Szekeres, George]]
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| | title = Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem
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| | journal = Acta Litt. Sci. Szeged
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| | volume = 7
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| | year = 1934
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| | pages = 95–102}}
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| * {{cite journal
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| | author = [[Solomon W. Golomb|Golomb, Solomon W.]]
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| | title = Powerful numbers
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| | journal = [[American Mathematical Monthly]]
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| | volume = 77
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| | issue = 8
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| | year = 1970
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| | pages = 848–852
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| | doi = 10.2307/2317020
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| | jstor = 2317020}}
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| * {{cite book
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| | author = [[Richard K. Guy|Guy, Richard K.]]
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| | pages = Section B16
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| | title = Unsolved Problems in Number Theory, 3rd edition
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| | publisher = Springer-Verlag
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| | year = 2004
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| | isbn = 0-387-20860-7
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| | nopp = true}}
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| * {{cite conference
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| | author = [[Roger Heath-Brown|Heath-Brown, Roger]]
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| | title = Ternary quadratic forms and sums of three square-full numbers
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| | booktitle = Séminaire de Théorie des Nombres, Paris, 1986-7
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| | publisher = Birkhäuser
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| | location = Boston
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| | pages = 137–163
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| | year = 1988}}
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| * {{cite conference
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| | author = Heath-Brown, Roger
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| | title = Sums of three square-full numbers
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| | booktitle = Number Theory, I (Budapest, 1987)
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| | publisher = Colloq. Math. Soc. János Bolyai, no. 51
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| | year = 1990
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| | pages = 163–171}}
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| * {{cite book | last=Ivić | first=Aleksandar | title=The Riemann zeta-function. The theory of the Riemann zeta-function with applications | series=A Wiley-Interscience Publication | location=New York etc. | publisher=John Wiley & Sons | year=1985 | isbn=0-471-80634-X | zbl=0556.10026 | pages=33–34,407–413 }}
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| * {{cite journal
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| | author = McDaniel, Wayne L.
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| | title = Representations of every integer as the difference of powerful numbers
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| | journal = [[Fibonacci Quarterly]]
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| | volume = 20
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| | year = 1982
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| | pages = 85–87}}
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| * {{cite journal
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| | author = Nitaj, Abderrahmane
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| | title = On a conjecture of Erdős on 3-powerful numbers
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| | journal = [[Bulletin of the London Mathematical Society|Bull. London Math. Soc.]]
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| | volume = 27
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| | year = 1995
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| | pages = 317–318
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| | doi = 10.1112/blms/27.4.317
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| | issue = 4}}
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| == External links ==
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| * {{mathworld|urlname=PowerfulNumber|title=Powerful number}}
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| * [http://www.math.unicaen.fr/~nitaj/abc.html The abc conjecture]
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| * {{SloanesRef |sequencenumber=A060355|name=Numbers n such that n and n+1 are a pair of consecutive powerful numbers}}
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| {{Divisor classes}}
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| {{Classes of natural numbers}}
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| [[Category:Integer sequences]]
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