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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.
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| ==Definition==
| | [http://www.Magisterdireccionteatral.Uchile.cl/contenido/birth-minecraft-wiki-enchanting uchile.cl]When I returned to retrieve my items, I discovered they had [http://www.aimlessly.info/2014/08/4-mistakes-in-minecraft-texture-packs-for-mac-that-make-you-look-dumb/ minecraft Seeds 1.7] all been consumed by flame, as well. If you might have not played Caravaneer before, or only touched on it briefly inside past, give it a go today and find out where your [http://Ecosoft.Com.mx/ecoforo/member.php?24542-HDech minecraft forums] trail leads. About how exactly i would personally provide this web site to let you know just a little inside the company. (Notch additionally shows them a person to buccaneer Minecraft. When [http://www.muaguide.com/getting-best-try-minecraft-free-ipad/ try minecraft demo] you hear the tell-tale hiss of a Creeper, [http://krakmh.ga/?author=347 try minecraft free online games] the first instinct could be to rush to the safety of the well-constructed base. Ghast's shoot fireballs at you which could blow up blocks that shelter you or that you are standing on.<br><br>To generate a personal server to meet your requirements and your buddys a regular personal pc is necessary, absolutely nothing also severe. You'll also place a mattress as part of your house, and sleep during the entire evening in order to avoid enemy mobs from breeding whatsoever.<br><br>Dye containing ([https://wpbuddy.zendesk.com/entries/43729455-Choosing-Minecraft-Texture-Packs-1-6-2 https://wpbuddy.zendesk.com]) your wooly target black, use as pictured, or mix to produce darker colour free minecraft. </a><img width="1" height="1" src="http://in. But how do you get rid of the Ender Dragon once you've taken your fated journey to The End. The End is probably going to become the final part on the journey of finishing free Minecraft.<br><br>She assists make-up artists, hair artists, hairstylists and beauty salon owners create an excellent lifestyle pursuing their fire inside the beauty salon industry. Firstly, consider your limitations, as towards the things you can create. Start with the first chapter Basic free Minecraft Setup or jump ahead while using navigation menu around the left.<br><br>Although the open-world sandbox game may not exactly are the best-looking software product around the market, it is going to feature better texture on Sony's new console. I just had two servers in thoughts that have been really good. |
| 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>.
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| ==Applications==
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| 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]].)
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| 5-smooth or [[regular number]]s play a special role in [[Babylonian mathematics]].<ref>{{citation
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| | authorlink = Aaboe
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| | last = Aaboe | first = Asger
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| | title = Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)
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| | journal = Journal of Cuneiform Studies
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| | volume = 19
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| | issue = 3
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| | pages = 79–86
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| | year = 1965
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| | id = {{MathSciNet | id = 0191779}}
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| | doi = 10.2307/1359089}}.</ref> They are also important in [[music theory]],<ref>{{citation
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| | last = Longuet-Higgins | first = H. C.
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| | year = 1962
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| | title = Letter to a musical friend
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| | journal = Music Review
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| | issue = August
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| | 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
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| | authorlink = Edsger W. Dijkstra | last = Dijkstra | first = Edsger W. | |
| | title = Hamming's exercise in SASL
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| | year = 1981
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| | url = http://www.cs.utexas.edu/users/EWD/ewd07xx/EWD792.PDF
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| | id = Report EWD792. Originally a privately circulated handwitten note}}.</ref>
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| 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]].
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| ==Distribution==
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| Let <math>\scriptstyle \Psi(x,y)</math> denote the number of ''y''-smooth integers less than or equal to ''x'' (the de Bruijn function).
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| If the smoothness bound ''B'' is fixed and small, there is a good estimate for <math>\scriptstyle\Psi(x,B)</math>:
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| :<math> \Psi(x,B) \sim \frac{1}{\pi(B)!} \prod_{p\le B}\frac{\log x}{\log p}. </math>
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| where <math>\scriptstyle{\pi(B)}</math> denotes the number of primes less than or equal to <math>\scriptstyle B</math>.
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| Otherwise, define the parameter ''u'' as ''u'' = log ''x'' / log ''y'': that is, ''x'' = ''y''<sup>''u''</sup>. Then,
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| :<math> \Psi(x,y) = x\cdot \rho(u) + O\left(\frac{x}{\log y}\right)</math>
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| where <math>\scriptstyle\rho(u)</math> is the [[Dickman function]].
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| ==Powersmooth numbers==<!-- This section is linked from [[Table of prime factors]] -->
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| Further, ''m'' is called ''B''-'''powersmooth''' (or ''B''-'''ultrafriable''') if all prime ''powers'' <math>\scriptstyle p^{\nu}</math> dividing ''m'' satisfy:
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| :<math>p^{\nu} \leq B.\,</math>
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| 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.
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| ''B''-smooth and ''B''-powersmooth numbers have applications in number theory, such as in [[Pollard's p − 1 algorithm|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 logarithm]]s has a running time of [[asymptotic notation|O]](''B''<sup>1/2</sup>) for [[group (mathematics)|group]]s of ''B''-smooth order.
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| ==See also==
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| *[[Rough number]]
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| *[[Størmer's theorem]]
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| *[[Highly composite number]]
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| ==Notes==
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| <references/> | |
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| ==References== | |
| * G. Tenenbaum, ''Introduction to analytic and probabilistic number theory'', (CUP, 1995) ISBN 0-521-41261-7
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| * [[A. Granville]], [http://www.dms.umontreal.ca/~andrew/PDF/msrire.pdf ''Smooth numbers: Computational number theory and beyond''], Proc. of MSRI workshop, 2008
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| ==External links==
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| * {{mathworld|urlname=SmoothNumber|title=Smooth Number}}
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| The [[On-Line Encyclopedia of Integer Sequences]] (OEIS) | |
| lists ''B''-smooth numbers for small ''B''s:
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| * 2-smooth numbers: [[OEIS:A000079|A000079]] (2<sup>''i''</sup>)
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| * 3-smooth numbers: [[OEIS:A003586|A003586]] (2<sup>''i''</sup>3<sup>''j''</sup>)
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| * 5-smooth numbers: [[OEIS:A051037|A051037]] (2<sup>''i''</sup>3<sup>''j''</sup>5<sup>''k''</sup>)
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| * 7-smooth numbers: [[OEIS:A002473|A002473]] (2<sup>''i''</sup>3<sup>''j''</sup>5<sup>''k''</sup>7<sup>''l''</sup>)
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| * 11-smooth numbers: [[OEIS:A051038|A051038]] (etc...)
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| * 13-smooth numbers: [[OEIS:A080197|A080197]]
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| * 17-smooth numbers: [[OEIS:A080681|A080681]]
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| * 19-smooth numbers: [[OEIS:A080682|A080682]]
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| * 23-smooth numbers: [[OEIS:A080683|A080683]]
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| {{Divisor classes}}
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| {{Classes of natural numbers}}
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| [[Category:Analytic number theory]]
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| [[Category:Integer sequences]]
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