Powerful number

A powerful number is a positive integer m such 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, where a and b are positive integers. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful. Powerful numbers are also known as squarefull, square-full, or 2-full.^[1] (Not to be confused with the term squareful, which refers to numbers that are not square-free.^{[2] [3]} )

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 the OEIS).

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

${\displaystyle m=\prod p_{i}^{\alpha _{i}},}${\displaystyle 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

${\displaystyle m=\left(\prod p_{i}^{\beta _{i}}\right)\left(\prod p_{i}^{\gamma _{i}}\right)=\left(\prod p_{i}^{\beta _{i}/2}\right)^{2}\left(\prod p_{i}^{\gamma _{i}/3}\right)^{3}}${\displaystyle m=\left(\prod p_{i}^{\beta _{i}}\right)\left(\prod p_{i}^{\gamma _{i}}\right)=\left(\prod p_{i}^{\beta _{i}/2}\right)^{2}\left(\prod p_{i}^{\gamma _{i}/3}\right)^{3}}

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

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/b³ is an integer. In addition, each prime factor of m/b³ has an even exponent, so m/b³ is a perfect square, so call this a²; then m = a²b³. For example:

${\displaystyle m=21600=2^{5}\times 3^{3}\times 5^{2},,円}${\displaystyle m=21600=2^{5}\times 3^{3}\times 5^{2},,円}

${\displaystyle b=2\times 3=6,,円}${\displaystyle b=2\times 3=6,,円}

${\displaystyle a={\sqrt {\frac {m}{b^{3}}}}={\sqrt {2^{2}\times 5^{2}}}=10,,円}${\displaystyle a={\sqrt {\frac {m}{b^{3}}}}={\sqrt {2^{2}\times 5^{2}}}=10,,円}

${\displaystyle m=a^{2}b^{3}=10^{2}\times 6^{3},円.}${\displaystyle m=a^{2}b^{3}=10^{2}\times 6^{3},円.}

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

The sum of the reciprocals of the powerful numbers converges. The value of this sum may be written in several other ways, including as the infinite product

${\displaystyle \prod _{p}\left(1+{\frac {1}{p(p-1)}}\right)={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}={\frac {315}{2\pi ^{4}}}\zeta (3)=1.9435964368\ldots ,}${\displaystyle \prod _{p}\left(1+{\frac {1}{p(p-1)}}\right)={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}={\frac {315}{2\pi ^{4}}}\zeta (3)=1.9435964368\ldots ,}

where p runs over all primes, ζ(s) denotes the Riemann zeta function, and ζ(3) is Apéry's constant.^[4] (sequence A082695 in the OEIS) More generally, the sum of the reciprocals of the sth powers of the powerful numbers (a Dirichlet series generating function) is equal to

${\displaystyle {\frac {\zeta (2s)\zeta (3s)}{\zeta (6s)}}}${\displaystyle {\frac {\zeta (2s)\zeta (3s)}{\zeta (6s)}}}

whenever it converges.

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,

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

(Golomb, 1970).

The two smallest consecutive powerful numbers are 8 and 9. Since Pell's equation ${\displaystyle x^{2}-8y^{2}=1}${\displaystyle x^{2}-8y^{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 ${\displaystyle x^{2}-ny^{2}=\pm 1}${\displaystyle x^{2}-ny^{2}=\pm 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 ${\displaystyle (23^{3},2^{3}3^{2}13^{2})}${\displaystyle (23^{3},2^{3}3^{2}13^{2})} in which neither number in the pair is a square. Walker (1976) showed that there are indeed infinitely many such pairs by showing that ${\displaystyle 3^{3}c^{2}+1=7^{3}d^{2}}${\displaystyle 3^{3}c^{2}+1=7^{3}d^{2}} has infinitely many solutions. Walker's solutions to this equation are generated, for any odd integer ${\displaystyle k}${\displaystyle k}, by considering the number

${\displaystyle (2{\sqrt {7}}+3{\sqrt {3}})^{7k}=a{\sqrt {7}}+b{\sqrt {3}},}${\displaystyle (2{\sqrt {7}}+3{\sqrt {3}})^{7k}=a{\sqrt {7}}+b{\sqrt {3}},}

for integers ${\displaystyle a}${\displaystyle a} divisible by 7 and ${\displaystyle b}${\displaystyle b} divisible by 3, and constructing from ${\displaystyle a}${\displaystyle a} and ${\displaystyle b}${\displaystyle b} the consecutive powerful numbers ${\displaystyle 7a^{2}}${\displaystyle 7a^{2}} and ${\displaystyle 3b^{2}}${\displaystyle 3b^{2}} with ${\displaystyle 7a^{2}=1+3b^{2}}${\displaystyle 7a^{2}=1+3b^{2}}. The smallest consecutive pair in this family is generated for ${\displaystyle k=1}${\displaystyle k=1}, ${\displaystyle a=2637362}${\displaystyle a=2637362}, and ${\displaystyle b=4028637}${\displaystyle b=4028637} as

${\displaystyle 7\cdot 2637362^{2}=2^{2}\cdot 7^{3}\cdot 13^{2}\cdot 43^{2}\cdot 337^{2}=48689748233308}${\displaystyle 7\cdot 2637362^{2}=2^{2}\cdot 7^{3}\cdot 13^{2}\cdot 43^{2}\cdot 337^{2}=48689748233308}

and

${\displaystyle 3\cdot 4028637^{2}=3^{3}\cdot 139^{2}\cdot 9661^{2}=48689748233307.}${\displaystyle 3\cdot 4028637^{2}=3^{3}\cdot 139^{2}\cdot 9661^{2}=48689748233307.}

It is a conjecture of Erdős, Mollin, and Walsh that there are no three consecutive powerful numbers. If a triplet of consecutive powerful numbers exists, then its smallest term must be congruent to 7, 27, or 35 modulo 36.^[5]

If the abc conjecture is true, there are only a finite number of sets of three consecutive powerful numbers.

Any odd number is a difference of two consecutive squares: (k + 1)² = k² + 2k + 1, so (k + 1)² − k² = 2k + 1. Similarly, any multiple of four is a difference of the squares of two numbers that differ by two: (k + 2)² − k² = 4k + 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:

2 = 3³ − 5²

10 = 13³ − 3⁷

18 = 19² − 7³ = 3⁵ − 15².

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).

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.

• Achilles number
• Highly powerful number

• Cohn, J. H. E. (1998). "A conjecture of Erdős on 3-powerful numbers". Math. Comp. 67 (221): 439–440. doi:10.1090/S0025-5718-98-00881-3 .
• Erdős, Paul & 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 (8): 848–852. doi:10.2307/2317020. JSTOR 2317020.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. Section B16. ISBN 978-0-387-20860-2.
• Heath-Brown, Roger (1988). "Ternary quadratic forms and sums of three square-full numbers". Séminaire de Théorie des Nombres, Paris, 1986-7. Boston: Birkhäuser. pp. 137–163.
• Heath-Brown, Roger (1990). "Sums of three square-full numbers". Number Theory, I (Budapest, 1987). Colloq. Math. Soc. János Bolyai, no. 51. pp. 163–171.
• Ivić, Aleksandar (1985). The Riemann zeta-function. The theory of the Riemann zeta-function with applications. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. pp. 33–34, 407–413. ISBN 978-0-471-80634-9. Zbl 0556.10026.
• McDaniel, Wayne L. (1982). "Representations of every integer as the difference of powerful numbers". Fibonacci Quarterly . 20: 85–87. doi:10.1080/00150517.1982.12430037.
• Nitaj, Abderrahmane (1995). "On a conjecture of Erdős on 3-powerful numbers". Bull. London Math. Soc. 27 (4): 317–318. CiteSeerX 10.1.1.24.563 . doi:10.1112/blms/27.4.317.
• Walker, David T. (1976). "Consecutive integer pairs of powerful numbers and related Diophantine equations" (PDF). The Fibonacci Quarterly. 14 (2): 111–116. doi:10.1080/00150517.1976.12430562. MR 0409348.

• Power-full number at Encyclopedia of Mathematics.
• Weisstein, Eric W. "Powerful number". MathWorld .
• The abc conjecture
• OEIS sequence A060355 (Numbers n such that n and n+1 are a pair of consecutive powerful numbers)

• Integer sequences
• Abc conjecture

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