Not What You Meant? There are 15 definitions for Erdős conjecture.

Powerful number

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Divisibility-based
sets of integers

Form of factorization:

Powerful number

Constrained divisor sums:

Perfect number

Almost perfect number

Quasiperfect number

Multiply perfect number

Hyperperfect number

Unitary perfect number

Semiperfect number

Primitive semiperfect number

Practical number

Numbers with many divisors:

Abundant number

Highly abundant number

Superabundant number

Colossally abundant number

Highly composite number

Superior highly composite number

Other:

Harmonic divisor number

Frugal number

Equidigital number

Extravagant number

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

<math>m = \prod p_i^{\alpha_i},</math>

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

<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>

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.

Mathematical properties

The sum of reciprocals of powerful numbers converges to

<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>

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,

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

(Golomb, 1970). The sequence of pairs of consecutive powerful numbers is given by A060355. 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); more generally, one can find consecutive powerful numbers by solving a similar Pell equation x² − ny² = ±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. 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.

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).

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.

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.
McDaniel, Wayne L. (1982). "Representations of every integer as the difference of powerful numbers". Fibonacci Quarterly 20: 85–87.
Nitaj, Abderrahmane (1995). "On a conjecture of Erdős on 3-powerful numbers". Bull. London Math. Soc. 27: 317–318.