Smooth number

A positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 2² × 3⁴ × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, respectively. All 5-smooth numbers are of the form 2^a × 3^b × 5^c, where a, b and c are non-negative integers.

The 3-smooth numbers have also been called "harmonic numbers", although that name has other more widely used meanings.[4] 5-smooth numbers are also called regular numbers or Hamming numbers;[5] 7-smooth numbers are also called humble numbers,[6] and sometimes called highly composite,[7] although this conflicts with another meaning of highly composite numbers.

Here, note that B itself is not required to appear among the factors of a B-smooth number. If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. In many scenarios B is prime, but composite numbers are permitted as well. A number is B-smooth if and only if it is p-smooth, where p is the largest prime less than or equal to B.

An important practical application of smooth numbers is the fast Fourier transform (FFT) algorithms (such as the Cooley–Tukey FFT algorithm), which operates by recursively breaking down a problem of a given size n into problems the size of its factors. By using B-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. (Large prime sizes require less-efficient algorithms such as Bluestein's FFT algorithm.)

5-smooth or regular numbers play a special role in Babylonian mathematics.[8] They are also important in music theory (see Limit (music)),[9] and the problem of generating these numbers efficiently has been used as a test problem for functional programming.[10]

Smooth numbers have a number of applications to cryptography.[11] While most applications center around cryptanalysis (e.g. the fastest known integer factorization algorithms, for example: General number field sieve algorithm), the VSH hash function is another example of a constructive use of smoothness to obtain a provably secure design.

Let ${\displaystyle \Psi (x,y)}$ denote the number of y-smooth integers less than or equal to x (the de Bruijn function).

If the smoothness bound B is fixed and small, there is a good estimate for ${\displaystyle \Psi (x,B)}$:

${\displaystyle \Psi (x,B)\sim {\frac {1}{\pi (B)!}}\prod _{p\leq B}{\frac {\log x}{\log p}}.}$

where ${\displaystyle \pi (B)}$ denotes the number of primes less than or equal to ${\displaystyle B}$.

Otherwise, define the parameter u as u = log x / log y: that is, x = y^u. Then,

${\displaystyle \Psi (x,y)=x\cdot \rho (u)+O\left({\frac {x}{\log y}}\right)}$

where ${\displaystyle \rho (u)}$ is the Dickman function.

The average size of the smooth part of a number of given size is known as ${\displaystyle \zeta (u)}$, and it is known to decay much more slowly than ${\displaystyle \rho (u)}$.[12]

For any k, almost all natural numbers will not be k-smooth.

Further, m is called B-powersmooth (or B-ultrafriable) if all prime powers ${\displaystyle p^{\nu }}$ dividing m satisfy:

${\displaystyle p^{\nu }\leq B.\,}$

For example, 720 (2⁴ × 3² × 5¹) is 5-smooth but not 5-powersmooth (because there are several prime powers greater than 5, e.g. ${\displaystyle 3^{2}=9\nleq 5}$ and ${\displaystyle 2^{4}=16\nleq 5}$). It is 16-powersmooth since its greatest prime factor power is 2⁴ = 16. The number is also 17-powersmooth, 18-powersmooth, etc.

B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollard's p − 1 algorithm and ECM. Such applications are often said to work with "smooth numbers," with no B specified; this means the numbers involved must be B-powersmooth, for some unspecified small number B. As B increases, the performance of the algorithm or method in question degrades rapidly. For example, the Pohlig–Hellman algorithm for computing discrete logarithms has a running time of O(B^1/2)—for groups of B-smooth order.

Moreover, m is said to be smooth over a set A if there exists a factorization of m where the factors are powers of elements in A. For example, since 12 = 4 × 3, 12 is smooth over the sets A₁ = {4, 3}, A₂ = {2, 3}, and ${\displaystyle \mathbb {Z} }$, however it would not be smooth over the set A₃ = {3, 5}, as 12 contains the factor 4 = 2², which is not in A₃.

Note the set A does not have to be a set of prime factors, but it is typically a proper subset of the primes as seen in the factor base of Dixon's factorization method and the quadratic sieve. Likewise, it is what the general number field sieve uses to build its notion of smoothness, under the homomorphism ${\displaystyle \phi$ :\mathbb {Z} [\theta ]\to \mathbb {Z} /n\mathbb {Z} } .[13]

• Highly composite number
• Rough number
• Round number
• Størmer's theorem
• Unusual number

• G. Tenenbaum, Introduction to analytic and probabilistic number theory, (AMS, 2015) ISBN 978-0821898543
• A. Granville, Smooth numbers: Computational number theory and beyond, Proc. of MSRI workshop, 2008

• Weisstein, Eric W. "Smooth Number". MathWorld.

The On-Line Encyclopedia of Integer Sequences (OEIS) lists B-smooth numbers for small Bs:

• 2-smooth numbers: A000079 (2ⁱ)
• 3-smooth numbers: A003586 (2ⁱ3^j)
• 5-smooth numbers: A051037 (2ⁱ3^j5^k)
• 7-smooth numbers: A002473 (2ⁱ3^j5^k7^l)
• 11-smooth numbers: A051038 (etc...)
• 13-smooth numbers: A080197
• 17-smooth numbers: A080681
• 19-smooth numbers: A080682
• 23-smooth numbers: A080683