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In [[arithmetic]], an [[Odd number|odd]] [[composite number|composite]] [[integer]] ''n'' is called an '''Euler pseudoprime''' to base ''a'', if ''a'' and ''n'' are [[coprime]], and
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


: <math>a^{(n-1)/2} \equiv \pm 1\pmod{n}</math>
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.


(where ''mod'' refers to the [[modular arithmetic|modulo]] operation).
Registered users will be able to choose between the following three rendering modes:


The motivation for this definition is the fact that all [[prime number]]s ''p'' satisfy the above equation which can be deduced from [[Fermat's little theorem]]. Fermat's theorem  asserts that if ''p'' is prime, and coprime to ''a'', then ''a''<sup>''p''&minus;1</sup> =  1 (mod ''p''). Suppose that ''p''>2 is prime, then ''p'' can be expressed as 2''q''&nbsp;+&nbsp;1 where ''q'' is an integer. Thus; ''a''<sup>(2''q''+1)&nbsp;&minus;&nbsp;1</sup> = 1 (mod&nbsp;''p'') which means that ''a''<sup>2''q''</sup>&nbsp;&minus;&nbsp;1 = 0 (mod ''p''). This can be factored as (''a''<sup>''q''</sup>&nbsp;&minus;&nbsp;1)(''a''<sup>''q''</sup> + 1) = 0 (mod ''p'') which is equivalent to ''a''<sup>(''p''&minus;1)/2</sup> = ±1 (mod&nbsp;''p'').
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


The equation can be tested rather quickly, which can be used for probabilistic [[prime testing|primality testing]]. These tests are twice as strong as tests based on Fermat's little theorem.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


Every Euler pseudoprime is also a Fermat [[pseudoprime]]. It is not possible to produce a definite test of primality based on whether a [[number]] is an Euler pseudoprime because there exist ''absolute Euler pseudoprimes'', numbers which are Euler pseudoprimes to every base relatively prime to themselves. The absolute Euler pseudoprimes are a [[subset]] of the absolute Fermat pseudoprimes, or [[Carmichael number]]s, and the smallest absolute Euler pseudoprime is [[1729 (number)|1729]] = 7&times;13&times;19.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The stronger condition that ''a''<sup>(''n''&minus;1)/2</sup> =  (''a''/''n'') (mod ''n''), where (''a'',''n'') = 1 and (''a''/''n'') is the [[Jacobi symbol]], is sometimes used for a definition of an Euler pseudoprime. A discussion of numbers of this form can be found at [[Euler&ndash;Jacobi pseudoprime]].
<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].


==See also==
==Demos==
* [[Probable prime]]


==References==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


*N. Koblitz, "A Course in Number Theory and Cryptography", Springer-Verlag, 1987.
*H. Riesel, "Prime numbers and computer methods of factorisation", Birkhäuser, Boston, Mass., 1985.


{{DEFAULTSORT:Euler Pseudoprime}}
* accessibility:
[[Category:Pseudoprimes]]
** 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.


[[de:Eulersche Pseudoprimzahl]]
==Test pages ==
[[fr:Nombre pseudopremier d'Euler]]
 
[[it:Pseudoprimo di Eulero]]
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==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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