Powerful number

A powerful number is a positive integer "m" that for every prime number "p" dividing "m", "p"² also divides "m". Equivalently, a powerful number is the product of a square and a cube, that is, a number "m" of the form "m" = "a"²"b"³, 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".

The following is a list of all powerful numbers between 1 and 1000::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.

Equivalence of the two definitions

If "m" = "a"²"b"³, 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.

In the other direction, suppose that "m" is powerful, with prime factorization:$m\; =\; prod\; p\_i^\{alpha\_i\},$where each α_i ≥ 2. Define γ_i to be three if α_i is odd, and zero otherwise, and define β_i = α_i - γ_i. Then, all values β_i are nonnegative even integers, and all values γ_i are either zero or three, so:supplies the desired representation of "m" as a product of a square and a cube.

The representation "m" = "a"²"b"³ calculated in this way has the property that "b" is squarefree, and is uniquely defined by this property.

Mathematical properties

The sum of reciprocals of powerful numbers converges to

:$prod\_pleft(1+frac\{1\}\{p(p-1)\}\; ight)=frac\{zeta(2)zeta(3)\}\{zeta(6)\}\; =\; frac\{315\}\{2pi^4\}zeta(3),$

where "p" runs over all primes, ζ("s") denotes the Riemann zeta function, and ζ(3) is Apéry's constant (Golomb, 1970).

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,

:$cx^\{1/2\}-3x^\{1/3\}le\; k(x)\; le\; cx^\{1/2\},\; c=zeta(3/2)/zeta(3)=2.173cdots$

(Golomb, 1970).

The sequence of pairs of consecutive powerful numbers is given by OEIS2C|id=A060355. The two smallest consecutive powerful numbers are 8 and 9. Since Pell's equation "x"² − 8"y"² = 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"² − "ny"² = ±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 (23³, 2³3²13²) in which neither number in the pair is a square. Jaroslaw Wroblewski showed that there are indeed infinitely many such pairs by showing that 3³c²+1=7³d² has infinitely many solutions. It is a conjecture of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers.

Sums and differences of powerful numbers

Any odd number is a difference of two consecutive squares: 2"k" + 1 = ("k" + 1)² - "k"². Similarly, any multiple of four is a difference of the squares of two numbers that differ by two. 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:

:2 = 3³ − 5²:10 = 13³ − 3⁷:18 = 19² − 7³ = 3²(3³ − 5²).

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

:6 = 5⁴7³ − 463²,

and McDaniel showed that every integer has infinitely many such representations(McDaniel, 1982).

Erds conjectured that every sufficiently large integer is a sum of at most three powerful numbers; this was proved by Roger Heath-Brown (1987).

Generalization

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.

:(2^"k"+1 − 1)^"k", 2^"k"(2^"k"+1 − 1)^"k", (2^"k"+1 − 1)^"k"+1

are "k"-powerful numbers in an arithmetic progression. Moreover, if "a"₁, "a"₂, ..., "a"_"s" are "k"-powerful in an arithmetic progression with common difference "d", then

: "a"₁("a"_"s" + "d")^"k", "a"₂("a"_"s" + "d")^"k", ..., "a"_"s"("a"_"s" + "d")^"k", ("a"_"s" + d)^"k"+1

are "s" + 1 "k"-powerful numbers in an arithmetic progression.

We have an identity involving "k"-powerful numbers:

:"a"^"k"("a"^"l" + ... + 1)^"k" + "a"^{"k" + 1}("a"^"l" + ... + 1)^"k" + ... + "a"^{"k" + "l"}("a"^"l" + ... + 1)^"k" = "a"^"k"("a"^"l" + ... +1)^"k"+1.

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

:"X" = 9712247684771506604963490444281, "Y" = 32295800804958334401937923416351, "Z" = 27474621855216870941749052236511

is a solution of the equation 32"X"³ + 49"Y"³ = 81"Z"³. We can construct another solution by setting "X"′ = "X"(49"Y"³ + 81"Z"³), "Y"′ = −"Y"(32"X"³ + 81"Z"³), "Z"′ = "Z"(32"X"³ − 49"Y"³) and omitting the common divisor.

See also

*Achilles number

References

* cite journal
author = Cohn, J. H. E.
title = A conjecture of Erdős on 3-powerful numbers
journal = Math. Comp.
volume = 67
year = 1998
pages = 439–440
url = http://www.ams.org/mcom/1998-67-221/S0025-5718-98-00881-3/ | doi = 10.1090/S0025-5718-98-00881-3

* cite journal
author = Erdős, Paul and Szekeres, George
title = Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem
journal = Acta Litt. Sci. Szeged
volume = 7
year = 1934
pages = 95–102

* cite journal
author = Golomb, Solomon W.
title = Powerful numbers
journal = American Mathematical Monthly
volume = 77
year = 1970
pages = 848–852
doi = 10.2307/2317020

* cite book
author = Guy, Richard K.
pages = Section B16
title = Unsolved Problems in Number Theory, 3rd edition
publisher = Springer-Verlag
year = 2004
id = ISBN 0-387-20860-7

* cite conference
author = Heath-Brown, Roger
title = Ternary quadratic forms and sums of three square-full numbers
booktitle = Séminaire de Théorie des Nombres, Paris, 1986-7
publisher = Birkhäuser
location = Boston
pages = 137–163
year = 1988

* cite conference
author = Heath-Brown, Roger
title = Sums of three square-full numbers
booktitle = Number Theory, I (Budapest, 1987)
publisher = Colloq. Math. Soc. János Bolyai, no. 51
year = 1990
pages = 163–171

* cite journal
author = McDaniel, Wayne L.
title = Representations of every integer as the difference of powerful numbers
journal = Fibonacci Quarterly
volume = 20
year = 1982
pages = 85–87

* cite journal
author = Nitaj, Abderrahmane
title = On a conjecture of Erdős on 3-powerful numbers
journal = Bull. London Math. Soc.
volume = 27
year = 1995
pages = 317–318
doi = 10.1112/blms/27.4.317

External links

*
* [http://www.math.unicaen.fr/~nitaj/abc.html The abc conjecture]