Powerful number
Encyclopedia
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. 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:
In the other direction, suppose that m is powerful, with prime factorization
where each αi ≥ 2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi - γi. Then, all values βi are nonnegative even integers, and all values γi are either zero or three, so
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:
The representation m = a2b3 calculated in this way has the property that b is squarefree
, and is uniquely defined by this property.
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,
(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); 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. 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.
, 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:
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
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).
k, 2k(2k+1 − 1)k, (2k+1 − 1)k+1
are k-powerful numbers in an arithmetic progression
. Moreover, if a1, a2, ..., as are k-powerful in an arithmetic progression with common difference d, then
a2(as + d)k, ..., as(as + d)k, (as + d)k+1
are s + 1 k-powerful numbers in an arithmetic progression.
We have an identity involving k-powerful numbers:
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
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.
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
and a cube
Cube (arithmetic)
In arithmetic and algebra, the cube of a number n is its third power — the result of the number multiplying by itself three times:...
, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
and George Szekeres
George Szekeres
George Szekeres AM was a Hungarian-Australian mathematician.-Early years:Szekeres was born in Budapest, Hungary as Szekeres György and received his degree in chemistry at the Technical University of Budapest. He worked six years in Budapest as an analytical chemist. He married Esther Klein in 1936...
studied such numbers and Solomon W. Golomb
Solomon W. Golomb
Solomon Wolf Golomb is an American mathematician and engineer and a professor of electrical engineering at the University of Southern California, best known to the general public and fans of mathematical games as the inventor of polyominoes, the inspiration for the computer game Tetris...
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 .
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
where each αi ≥ 2. Define γi to be three if αi is odd, and zero otherwise, and define βi = αi - γi. Then, all values βi are nonnegative even integers, and all values γi are either zero or three, so
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:
The representation m = a2b3 calculated in this way has the property that b is squarefree
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
, and is uniquely defined by this property.
Mathematical properties
The sum of reciprocals of powerful numbers converges towhere p runs over all primes, ζ(s) denotes the Riemann zeta function, and ζ(3) is Apéry's constant
Apéry's constant
In mathematics, Apéry's constant is a number that occurs in a variety of situations. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics...
(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
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
of x. More precisely,
(Golomb, 1970).
The two smallest consecutive powerful numbers are 8 and 9. Since Pell's equation
Pell's equation
Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this 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. 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
Erdos conjecture
The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects.Some of these are the following:* The Cameron–Erdős conjecture on sum-free sets of integers, proved by Ben Green....
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: (k + 1)2 = k2 + 2k +12, 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 numberSingly even number
In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not...
, 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
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
conjectured that every sufficiently large integer is a sum of at most three powerful numbers; this was proved by Roger Heath-Brown
Roger Heath-Brown
David Rodney "Roger" Heath-Brown F.R.S. , is a British mathematician working in the field of analytic number theory. He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker...
(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.k, 2k(2k+1 − 1)k, (2k+1 − 1)k+1
are k-powerful numbers in an arithmetic progression
Arithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...
. Moreover, if a1, a2, ..., as are k-powerful in an arithmetic progression with common difference d, then
- a1(as + d)k,
a2(as + d)k, ..., as(as + d)k, (as + d)k+1
are s + 1 k-powerful numbers in an arithmetic progression.
We have an identity involving k-powerful numbers:
- ak(al + ... + 1)k + ak + 1(al + ... + 1)k + ... + ak + l(al + ... + 1)k = ak(al + ... +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.