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In [[mathematics]], a '''semiprime''' (also called '''biprime''' or '''2-[[almost prime]]''', or '''pq number''') is a [[natural number]] that is the product of two (not necessarily distinct) [[prime number]]s. The semiprimes less than 100 are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, and 95. {{OEIS|id=A001358}}.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


By definition, semiprime numbers have no composite factors other than themselves. For example, the number 26 is semiprime and its only factors are 1, 2, 13, and 26.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==Properties==
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The total number of [[prime factor]]s Ω(''n'') for a semiprime ''n'' is two, by definition. A semiprime is either a [[Square number|square]] of a prime or [[Square-free integer|square-free]]. The square of any prime number is a semiprime, so the largest known semiprime will always be the square of the [[largest known prime]], unless the factors of the semiprime are not known. It is conceivable, but unlikely, that a way could be found to prove a larger number is a semiprime without knowing the two factors.<ref>Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=Semiprime ''The Prime Glossary: semiprime''] at The [[Prime Pages]]. Retrieved on 2013-09-04.</ref> A composite <math>n</math> non-divisible by primes <math>\le \sqrt[3]{n}</math> is semiprime. Various methods, such as elliptic pseudo-curves and the Goldwasser-Kilian ECPP theorem have been used to create provable, unfactored semiprimes with hundreds of digits.<ref>{{cite web|last=Broadhurst|first=David|url=http://physics.open.ac.uk/~dbroadhu/cert/semgpch.gp|title=To prove that N is a semiprime|date=12 March 2005|accessdate=2013-09-04}}</ref> These are considered novelties, since their construction method might prove vulnerable to factorization, and because it is simpler to multiply two primes together.


For a semiprime ''n''&nbsp;=&nbsp;''pq'' the value of [[Euler's totient function]] (the number of positive integers less than or equal to ''n'' that are [[relatively prime]] to ''n'') is particularly simple when ''p'' and ''q'' are distinct:
'''MathML'''
: &phi;(''n'') = (''p'' &minus; 1) (''q'' &minus; 1) = ''p'' ''q'' &minus; (''p'' + ''q'') + 1 = ''n'' &minus; (''p'' + ''q'') + 1.
:<math forcemathmode="mathml">E=mc^2</math>
If otherwise ''p'' and ''q'' are the same,
: &phi;(''n'') = &phi;(''p''<sup>2</sup>) = (''p'' &minus; 1) ''p'' = ''p''<sup>2</sup> &minus; ''p'' = ''n'' &minus; ''p''.


The concept of the [[prime zeta function]] can be adapted to semiprimes, which defines constants like
<!--'''PNG'''  (currently default in production)
: <math>\sum_{\Omega(n)=2} \frac{1}{n^2} \approx 0.1407604</math> {{OEIS|A117543}}
:<math forcemathmode="png">E=mc^2</math>
: <math>\sum_{\Omega(n)=2} \frac{1}{n(n-1)} \approx 0.17105</math> {{OEIS|A152447}}
: <math>\sum_{\Omega(n)=2} \frac{\ln n}{n^2} \approx 0.28360</math> {{OEIS|A154928}}


==Applications==
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


Semiprimes are highly useful in the area of [[cryptography]] and [[number theory]], most notably in [[public key cryptography]], where they are used by [[RSA (algorithm)|RSA]] and [[pseudorandom number generator]]s such as [[Blum Blum Shub]]. These methods rely on the fact that finding two large primes and multiplying them together (resulting in a semiprime) is computationally simple, whereas [[integer factorization|finding the original factors]] appears to be difficult. In the [[RSA Factoring Challenge]], [[RSA Security]] offered prizes for the factoring of specific large semiprimes and several prizes were awarded. The most recent such challenge closed in 2007.<ref>[http://www.rsa.com/rsalabs/node.asp?id=2092 Information Security, Governance, Risk, and Compliance - EMC]. RSA. Retrieved on 2014-05-11.</ref> <!-- The original RSA Factoring Challenge, issued in 1991, was replaced in 2001 by the New RSA Factoring Challenge; it was the latter challenge that was withdrawn in 2007. -->
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


In practical cryptography, it is not sufficient to choose just any semiprime; a good number must evade a number of [[Integer factorization#Special-purpose|well-known special-purpose algorithms]] that can factor numbers of certain form. The factors ''p'' and ''q'' of ''n'' should both be very large, around the same order of magnitude as the square root of ''n''; this makes [[trial division]] and [[Pollard's rho algorithm]] impractical. At the same time they should not be too close together, or else the number can be quickly factored by [[Fermat's factorization method]]. The number may also be chosen so that none of ''p''&nbsp;&minus;&nbsp;1, ''p''&nbsp;+&nbsp;1, ''q''&nbsp;&minus;&nbsp;1, or ''q''&nbsp;+&nbsp;1 are [[smooth number]]s, protecting against [[Pollard's p - 1 algorithm|Pollard's ''p''&nbsp;&minus;&nbsp;1 algorithm]] or [[Williams' p + 1 algorithm|Williams' ''p''&nbsp;+&nbsp;1 algorithm]].  However, these checks cannot take future algorithms or secret algorithms into account, introducing the possibility that numbers in use today may be broken by special-purpose algorithms.
==Demos==


In 1974 the [[Arecibo message]] was sent with a radio signal aimed at a [[star cluster]]. It consisted of 1679 binary digits intended to be interpreted as a 23&times;73 [[bitmap]] image. The number 1679 = 23&times;73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==See also==
*[[Chen's theorem]]


==References==
* accessibility:
<references/>
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


== External links ==
==Test pages ==
* {{MathWorld|title=Semiprime|urlname=Semiprime}}


{{Divisor classes}}
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
{{Prime number classes}}
*[[Displaystyle]]
{{Classes of natural numbers}}
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


[[Category:Integer sequences]]
*[[Inputtypes|Inputtypes (private Wikis only)]]
[[Category:Prime numbers]]
*[[Url2Image|Url2Image (private Wikis only)]]
[[Category:Theory of cryptography]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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