Powerful number

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A powerful number is a positive integer m that for every prime number p dividing m, p² 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 = a²b³. 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. (sequence A001694 in OEIS)

1 Equivalence of the two definitions
2 Mathematical properties
3 Sums and differences of powerful numbers
4 Generalization
5 See also
6 References
7 External links

[edit] Equivalence of the two definitions

If m = a²b³, 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

$m = (\prod p_i^{\beta_i})(\prod p_i^{\gamma_i}) = (\prod p_i^{\beta_i/2})^2(\prod p_i^{\gamma_i/3})^3$

supplies the desired representation of m as a product of a square and a cube.

The representation m = a²b³ calculated in this way has the property that b is squarefree, and is uniquely defined by this property.

[edit] Mathematical properties

The sum of reciprocals of powerful numbers converges to

$\prod_p\left(1+\frac{1}{p(p-1)}\right)=\frac{\zeta(2)\zeta(3)}{\zeta(6)} = \frac{315}{2\pi^4}\zeta(3),$

where p runs over all primes, ζ(s) denotes Riemann's 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,

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

(Golomb, 1970).

The two smallest consecutive powerful numbers are 8 and 9. Since Pell's equation x² − 8y² = 1 has infinitely many integral solutions, there are infinitely many pairs of consecutive powerful numbers (Golomb, 1970). The sequence of pairs of consecutive numbers is given by (sequence A060355 in OEIS). It is a conjecture of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers; if the abc conjecture is true, it would follow that there are only finitely many triples of consecutive powerful numbers.

[edit] Sums and differences of powerful numbers

Any odd number is a difference of two consecutive squares: 2k + 1 = (k + 1)² - k². 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).

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

[edit] 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 a₁, a₂, ..., a_s are k-powerful in an arithmetic progression with common difference d, then

a₁(a_s + d)^k,

a₂(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 32X³ + 49Y³ = 81Z³. We can construct another solution by setting X′ = X(49Y³ + 81Z³), Y′ = −Y(32X³ + 81Z³), Z′ = Z(32X³ − 49Y³) and omitting the common divisor.

[edit] See also

Achilles number

[edit] References

Cohn, J. H. E. (1998). "A conjecture of Erdős on 3-powerful numbers". Math. Comp. 67: 439–440.

Erdős, Paul and Szekeres, George (1934). "Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem". Acta Litt. Sci. Szeged 7: 95–102.

Golomb, Solomon W. (1970). "Powerful numbers". American Mathematical Monthly 77: 848–852.

Guy, Richard K. (2004). Unsolved Problems in Number Theory, 3rd edition. Springer-Verlag, Section B16. ISBN 0-387-20860-7.

Heath-Brown, Roger (1988). "Ternary quadratic forms and sums of three square-full numbers". Séminaire de Théorie des Nombres, Paris, 1986-7, 137–163, Boston: Birkhäuser.

Heath-Brown, Roger (1990). "Sums of three square-full numbers". Number Theory, I (Budapest, 1987), 163–171, Colloq. Math. Soc. János Bolyai, no. 51.