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In number theory, a smooth (or friable) number is an integer which factors completely into small prime numbers. The term seems to have been coined by Leonard Adleman.^[1] Smooth numbers are especially important in cryptography relying on factorization.

Definition

A positive integer is called B-smooth if none of its prime factors is 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, despite the fact that they miss out prime factors 3 and 5 respectively. 5-smooth numbers are also called regular numbers or Hamming numbers; 7-smooth numbers are also called humble, and sometimes called highly composite[1], although this conflicts with another meaning of that term.

Note that B does not have to be a prime factor. If the largest prime factor of a number is p then the number is B-smooth for any B ≥ p. Usually B is given as a prime, but composite numbers work 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.

Applications

An important practical application of smooth numbers is for fast Fourier transform (FFT) algorithms such as the Cooley–Tukey FFT algorithm that operate 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.^[2] They are also important in music theory,^[3] (see Limit (music)) and the problem of generating these numbers efficiently has been used as a test problem for functional programming.^[4]

Smooth numbers have a number of applications to cryptography.^[5] Although most applications involve cryptanalysis (e.g. the fastest known integer factorization algorithms), the VSH hash function is one example of a constructive use of smoothness to obtain a provably secure design.

Distribution

Let $\scriptstyle \Psi (x,y)$ 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 $\scriptstyle \Psi (x,B)$ :

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

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

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

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

where $\scriptstyle \rho (u)$ is the Dickman function.

Powersmooth numbers

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

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

For example, 2⁴3²5¹ is 5-smooth, but is not 5-powersmooth. 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. Such applications are often said to work with "smooth numbers," with no B specified; this means the numbers involved must be B-smooth 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.

Notes

↑ M. E. Hellman, J. M. Reyneri, "Fast computation of discrete logarithms in GF (q)", in Advances in Cryptology: Proceedings of CRYPTO '82 (eds. D. Chaum, R. Rivest, A. Sherman), New York: Plenum Press, 1983, p. 3–13, online quote at Google Scholar: "Adleman refers to integers which factor completely into small primes as “smooth” numbers."
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
↑ David Naccache, Igor Shparlinski, "Divisibility, Smoothness and Cryptographic Applications", http://eprint.iacr.org/2008/437.pdf

References

G. Tenenbaum, Introduction to analytic and probabilistic number theory, (CUP, 1995) ISBN 0-521-41261-7
A. Granville, Smooth numbers: Computational number theory and beyond, Proc. of MSRI workshop, 2008

External links

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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

Template:Divisor classes Template:Classes of natural numbers

[1] M. E. Hellman, J. M. Reyneri, "Fast computation of discrete logarithms in GF (q)", in Advances in Cryptology: Proceedings of CRYPTO '82 (eds. D. Chaum, R. Rivest, A. Sherman), New York: Plenum Press, 1983, p. 3–13, online quote at Google Scholar: "Adleman refers to integers which factor completely into small primes as “smooth” numbers."

[2] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[3] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[4] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

[5] David Naccache, Igor Shparlinski, "Divisibility, Smoothness and Cryptographic Applications", http://eprint.iacr.org/2008/437.pdf

[1]

[2]

[3]

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+In [[number theory]], a '''smooth (or friable) number''' is an [[integer]] which [[integer factorization|factors]] completely into small [[prime number]]s.  The term seems to have been coined by [[Leonard Adleman]].<ref>M. E. Hellman, J. M. Reyneri, "Fast computation of discrete logarithms in GF (q)", in ''Advances in Cryptology: Proceedings of CRYPTO '82'' (eds. D. Chaum, R. Rivest, A. Sherman), New York: Plenum Press, 1983, p. 3–13, [http://scholar.google.com/scholar?q=%22Adleman+refers+to+integers+which+factor+completely+into+small+primes+as+smooth+numbers.%22 online quote] at [[Google Scholar]]: "Adleman refers to integers which factor completely into small primes as “smooth” numbers."</ref> Smooth numbers are especially important in [[cryptography]] relying on factorization.
+==Definition==
+A [[negative and positive numbers|positive]] [[integer]] is called <var>B</var>-'''smooth''' if none of its [[prime factor]]s is greater than <var>B</var>. For example, 1,620 has prime factorization 2<sup>2</sup> × 3<sup>4</sup> × 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, despite the fact that they miss out prime factors 3 and 5 respectively. 5-smooth numbers are also called ''[[regular number]]s'' or ''Hamming numbers''; 7-smooth numbers are also called ''humble'', and sometimes called ''highly composite''[http://oeis.org/search?q=humble+number&sort=&language=&go=Search], although this conflicts with [[highly composite number|another meaning of that term]].
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+Note that <var>B</var> does not have to be a prime factor. If the largest prime factor of a number is <var>p</var> then the number is <var>B</var>-smooth for any <var>B</var> ≥ <var>p</var>. Usually <var>B</var> is given as a prime, but [[composite number]]s work as well. A number is <var>B</var>-smooth [[if and only if]] it is <var>p</var>-smooth, where <var>p</var> is the largest prime less than or equal to <var>B</var>.
+==Applications==
+An important practical application of smooth numbers is for [[fast Fourier transform]] (FFT) algorithms such as the [[Cooley–Tukey FFT algorithm]] that operate 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]].)
+-smooth or [[regular number]]s play a special role in [[Babylonian mathematics]].<ref>{{citation
+ | authorlink = Aaboe
+ | last = Aaboe | first = Asger
+ | title = Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)
+ | journal = Journal of Cuneiform Studies
+ | volume = 19
+ | issue = 3
+ | pages = 79–86
+ | year = 1965
+ | id = {{MathSciNet | id = 0191779}}
+ | doi = 10.2307/1359089}}.</ref> They are also important in [[music theory]],<ref>{{citation
+  | last = Longuet-Higgins | first = H. C.
+ | year = 1962
+ | title = Letter to a musical friend
+ | journal = Music Review
+ | issue = August
+ | pages = 244–248}}.</ref> (see [[Limit (music)]]) and the problem of generating these numbers efficiently has been used as a test problem for [[functional programming]].<ref>{{citation
+ | authorlink = Edsger W. Dijkstra | last = Dijkstra | first = Edsger W.
+ | title = Hamming's exercise in SASL
+ | year = 1981
+ | url = http://www.cs.utexas.edu/users/EWD/ewd07xx/EWD792.PDF
+ | id = Report EWD792. Originally a privately circulated handwitten note}}.</ref>
+Smooth numbers have a number of applications to cryptography.<ref>David Naccache, Igor Shparlinski, "Divisibility, Smoothness and Cryptographic Applications", http://eprint.iacr.org/2008/437.pdf</ref>  Although most applications involve [[cryptanalysis]] (e.g. the fastest known [[integer factorization]] algorithms), the [[Very smooth hash|VSH]] hash function is one example of a constructive use of smoothness to obtain a [[Provably secure cryptographic hash function|provably secure design]].
+==Distribution==
+Let <math>\scriptstyle \Psi(x,y)</math> 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 <math>\scriptstyle\Psi(x,B)</math>:
+:<math> \Psi(x,B) \sim  \frac{1}{\pi(B)!} \prod_{p\le B}\frac{\log x}{\log p}. </math>
+where <math>\scriptstyle{\pi(B)}</math> denotes the number of primes less than or equal to <math>\scriptstyle B</math>.
+Otherwise, define the parameter ''u'' as ''u'' = log&nbsp;''x''&nbsp;/&nbsp;log&nbsp;''y'': that is, ''x'' = ''y''<sup>''u''</sup>.  Then,
+:<math> \Psi(x,y) = x\cdot \rho(u) + O\left(\frac{x}{\log y}\right)</math>
+where <math>\scriptstyle\rho(u)</math> is the [[Dickman function]].
+==Powersmooth numbers==<!-- This section is linked from [[Table of prime factors]] -->
+Further, ''m'' is called ''B''-'''powersmooth''' (or ''B''-'''ultrafriable''') if all prime ''powers'' <math>\scriptstyle p^{\nu}</math> dividing ''m'' satisfy:
+:<math>p^{\nu} \leq B.\,</math>
+For example, 2<sup>4</sup>3<sup>2</sup>5<sup>1</sup> is 5-smooth, but is not 5-powersmooth. It is 16-powersmooth since its greatest prime factor power is 2<sup>4</sup> = 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 &minus; 1 algorithm|Pollard's ''p''&nbsp;&minus;&nbsp;1 algorithm]]. Such applications are often said to work with "smooth numbers," with no ''B'' specified; this means the numbers involved must be ''B''-smooth 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 logarithm]]s has a running time of [[asymptotic notation|O]](''B''<sup>1/2</sup>) for [[group (mathematics)|group]]s of ''B''-smooth order.
+==See also==
+*[[Rough number]]
+*[[Størmer's theorem]]
+*[[Highly composite number]]
+==Notes==
+<references/>
+==References==
+* G. Tenenbaum, ''Introduction to analytic and probabilistic number theory'', (CUP, 1995) ISBN 0-521-41261-7
+* [[A. Granville]], [http://www.dms.umontreal.ca/~andrew/PDF/msrire.pdf ''Smooth numbers: Computational number theory and beyond''], Proc. of MSRI workshop, 2008
+==External links==
+* {{mathworld|urlname=SmoothNumber|title=Smooth Number}}
+The [[On-Line Encyclopedia of Integer Sequences]] (OEIS)
+lists ''B''-smooth numbers for small ''B''s:
+* 2-smooth numbers: [[OEIS:A000079|A000079]] (2<sup>''i''</sup>)
+* 3-smooth numbers: [[OEIS:A003586|A003586]] (2<sup>''i''</sup>3<sup>''j''</sup>)
+* 5-smooth numbers: [[OEIS:A051037|A051037]] (2<sup>''i''</sup>3<sup>''j''</sup>5<sup>''k''</sup>)
+* 7-smooth numbers: [[OEIS:A002473|A002473]] (2<sup>''i''</sup>3<sup>''j''</sup>5<sup>''k''</sup>7<sup>''l''</sup>)
+* 11-smooth numbers: [[OEIS:A051038|A051038]] (etc...)
+* 13-smooth numbers: [[OEIS:A080197|A080197]]
+* 17-smooth numbers: [[OEIS:A080681|A080681]]
+* 19-smooth numbers: [[OEIS:A080682|A080682]]
+* 23-smooth numbers: [[OEIS:A080683|A080683]]
+{{Divisor classes}}
+{{Classes of natural numbers}}
+[[Category:Analytic number theory]]
+[[Category:Integer sequences]]