https://en.formulasearchengine.com/index.php?title=Chain_(algebraic_topology)&feed=atom&action=historyChain (algebraic topology) - Revision history2024-03-28T19:27:11ZRevision history for this page on the wikiMediaWiki 1.42.0-wmf.5https://en.formulasearchengine.com/index.php?title=Chain_(algebraic_topology)&diff=226446&oldid=prev77.70.120.85 at 15:49, 9 February 20142014-02-09T15:49:57Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:49, 9 February 2014</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">In [[arithmetic]]</del>, <del style="font-weight: bold; text-decoration: none;">an [[Odd number|odd]] [[composite number|composite]] [[integer]] ''n'' is called an '''Euler pseudoprime''' </del>to <del style="font-weight: bold; text-decoration: none;">base ''a''</del>, <del style="font-weight: bold; text-decoration: none;">if ''a'' and ''n'' are [[coprime]]</del>, and </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">It allows you to get first hand information on how to handle hives, open frames and harvest the honey. Secondly</ins>, <ins style="font-weight: bold; text-decoration: none;">this condition may also occur due </ins>to <ins style="font-weight: bold; text-decoration: none;">some medications</ins>, <ins style="font-weight: bold; text-decoration: none;">for instance anti-diabetic drugs. If you want</ins>, <ins style="font-weight: bold; text-decoration: none;">you can then sneak up to each of the men by the window </ins>and <ins style="font-weight: bold; text-decoration: none;">use </ins>a <ins style="font-weight: bold; text-decoration: none;">stun gun or takedown </ins>to <ins style="font-weight: bold; text-decoration: none;">clear </ins>the <ins style="font-weight: bold; text-decoration: none;">room. A child might need doses of steroid to treat his episode of hives</ins>. <ins style="font-weight: bold; text-decoration: none;">If the issue </ins>is <ins style="font-weight: bold; text-decoration: none;">something </ins>that can be <ins style="font-weight: bold; text-decoration: none;">removed then you should try to take that action as soon as possible</ins>. <ins style="font-weight: bold; text-decoration: none;"><br><br>That</ins>'s <ins style="font-weight: bold; text-decoration: none;">not to say </ins>that <ins style="font-weight: bold; text-decoration: none;">it</ins>'<ins style="font-weight: bold; text-decoration: none;">s not out of the question, but it</ins>'<ins style="font-weight: bold; text-decoration: none;">s most likely something else. The Key and Values are where you will find the actual information which modifies functions. She was afraid to speak up for fear of getting into trouble</ins>, and <ins style="font-weight: bold; text-decoration: none;">did anything she could </ins>to <ins style="font-weight: bold; text-decoration: none;">avoid confrontation. If we look at a planetary system through the Hubble space telescope, we are seeing what it was like many light years ago. Once you have a hive, you will want to gather a few extra bits of equipment, like a veil, </ins>a <ins style="font-weight: bold; text-decoration: none;">smoker</ins>, <ins style="font-weight: bold; text-decoration: none;">and </ins>a <ins style="font-weight: bold; text-decoration: none;">bee feeder. </ins><<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">The natural means of reproduction for honey bees is called swarming. In purely financial terms, the direct cost to replace a dead colony </ins>(<ins style="font-weight: bold; text-decoration: none;">or "dead-out"</ins>) <ins style="font-weight: bold; text-decoration: none;">will be in the area of $90 for new bees</ins>. <ins style="font-weight: bold; text-decoration: none;">In fact</ins>, <ins style="font-weight: bold; text-decoration: none;">at least 20% of the population suffers from an outbreak of hives at any given time. 24 online New England Journal of Medicine offers hope to sufferers of chronic hives. There are several home remedies for hives which are easy and safe to </ins>be <ins style="font-weight: bold; text-decoration: none;">tried at home</ins>. <<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">3: A successive number of people use an urticaria treatment </ins>that <ins style="font-weight: bold; text-decoration: none;">works from </ins>a <ins style="font-weight: bold; text-decoration: none;">different principle than just interpreting the urticaria </ins>as a <ins style="font-weight: bold; text-decoration: none;">mere byproduct of an allergy. Introduce this unit by reading </ins>a <ins style="font-weight: bold; text-decoration: none;">story about bees </ins>to <ins style="font-weight: bold; text-decoration: none;">your preschoolers</ins>. <ins style="font-weight: bold; text-decoration: none;">If you </ins>can <ins style="font-weight: bold; text-decoration: none;">follow all the above procedures and take care of your skin in a proper way</ins>, <ins style="font-weight: bold; text-decoration: none;">you </ins>can <ins style="font-weight: bold; text-decoration: none;">always prevent yourself from this skin problem</ins>. <ins style="font-weight: bold; text-decoration: none;">The first type is known </ins>as <ins style="font-weight: bold; text-decoration: none;">acute urticaria, and the second type is recognized </ins>as <ins style="font-weight: bold; text-decoration: none;">chronic urticaria</ins>. <ins style="font-weight: bold; text-decoration: none;">Consume events encompass </ins>a <ins style="font-weight: bold; text-decoration: none;">diverse range </ins>of <ins style="font-weight: bold; text-decoration: none;">objectives</ins>, <ins style="font-weight: bold; text-decoration: none;">some of </ins>which are <ins style="font-weight: bold; text-decoration: none;">extremely difficult. <br><br>It's one of the best urticaria treatments that you can use for your hive problem</ins>. The <ins style="font-weight: bold; text-decoration: none;">five registry hives that maintain this information </ins>are<ins style="font-weight: bold; text-decoration: none;">;. It seems that new diseases </ins>and <ins style="font-weight: bold; text-decoration: none;">viruses are being discovered all </ins>the <ins style="font-weight: bold; text-decoration: none;">time. This </ins>is <ins style="font-weight: bold; text-decoration: none;">the hardest because it virtually guarantees that Alex will be facing strike teams in addition to base defenses</ins>. <ins style="font-weight: bold; text-decoration: none;">I then only included the people in my study </ins>that <ins style="font-weight: bold; text-decoration: none;">had broken out in hives when I gave them the "Abracadabra" treatment. </ins><<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">When they swarm</ins>, <ins style="font-weight: bold; text-decoration: none;">honey bees carry no additional food with them. Aside from that</ins>, <ins style="font-weight: bold; text-decoration: none;">you will probably have to deal with bees which is challenging once they are around you. The largest element to keeping </ins>a <ins style="font-weight: bold; text-decoration: none;">beehive </ins>is <ins style="font-weight: bold; text-decoration: none;">what </ins>the <ins style="font-weight: bold; text-decoration: none;">colony </ins>is <ins style="font-weight: bold; text-decoration: none;">actually housed in</ins>. <ins style="font-weight: bold; text-decoration: none;">His system comes with a 100% guarantee so it will be </ins>the <ins style="font-weight: bold; text-decoration: none;">smartest decision you make</ins>. The <ins style="font-weight: bold; text-decoration: none;">job is done by people who have skills </ins>to <ins style="font-weight: bold; text-decoration: none;">perform it efficiently.</ins><<ins style="font-weight: bold; text-decoration: none;">br</ins>><<ins style="font-weight: bold; text-decoration: none;">br</ins>><ins style="font-weight: bold; text-decoration: none;">Here's more information on [</ins>http://www.<ins style="font-weight: bold; text-decoration: none;">joygoldkind</ins>.<ins style="font-weight: bold; text-decoration: none;">info</ins>/ <ins style="font-weight: bold; text-decoration: none;">how to treat hives</ins>] <ins style="font-weight: bold; text-decoration: none;">take a look </ins>at <ins style="font-weight: bold; text-decoration: none;">our web</ins>-<ins style="font-weight: bold; text-decoration: none;">site</ins>.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">: <math></del>a<del style="font-weight: bold; text-decoration: none;">^{(n-1)/2} \equiv \pm 1\pmod{n}</math></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">(where ''mod'' refers </del>to the <del style="font-weight: bold; text-decoration: none;">[[modular arithmetic|modulo]] operation)</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The motivation for this definition </del>is <del style="font-weight: bold; text-decoration: none;">the fact </del>that <del style="font-weight: bold; text-decoration: none;">all [[prime number]]s ''p'' satisfy the above equation which </del>can be <del style="font-weight: bold; text-decoration: none;">deduced from [[Fermat's little theorem]]</del>. <del style="font-weight: bold; text-decoration: none;">Fermat</del>'s <del style="font-weight: bold; text-decoration: none;">theorem asserts </del>that <del style="font-weight: bold; text-decoration: none;">if </del>''<del style="font-weight: bold; text-decoration: none;">p'' is prime</del>, and <del style="font-weight: bold; text-decoration: none;">coprime </del>to <del style="font-weight: bold; text-decoration: none;">''</del>a<del style="font-weight: bold; text-decoration: none;">''</del>, <del style="font-weight: bold; text-decoration: none;">then ''</del>a<del style="font-weight: bold; text-decoration: none;">''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">''p''&minus;1</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>> <del style="font-weight: bold; text-decoration: none;">= 1 </del>(<del style="font-weight: bold; text-decoration: none;">mod ''p''</del>). <del style="font-weight: bold; text-decoration: none;">Suppose that ''p''>2 is prime</del>, <del style="font-weight: bold; text-decoration: none;">then ''p'' can </del>be <del style="font-weight: bold; text-decoration: none;">expressed as 2''q''&nbsp;+&nbsp;1 where ''q'' is an integer</del>. <del style="font-weight: bold; text-decoration: none;">Thus; ''a''</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">(2''q''+1)&nbsp;&minus;&nbsp;1</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>> <del style="font-weight: bold; text-decoration: none;">= 1 (mod&nbsp;''p'') which means </del>that <del style="font-weight: bold; text-decoration: none;">''</del>a<del style="font-weight: bold; text-decoration: none;">''<sup>2''q''</sup>&nbsp;&minus;&nbsp;1 = 0 (mod ''p''). This can be factored </del>as <del style="font-weight: bold; text-decoration: none;">(''</del>a<del style="font-weight: bold; text-decoration: none;">''<sup>''q''</sup>&nbsp;&minus;&nbsp;1)(''</del>a<del style="font-weight: bold; text-decoration: none;">''<sup>''q''</sup> + 1) = 0 (mod ''p'') which is equivalent </del>to <del style="font-weight: bold; text-decoration: none;">''a''<sup>(''p''&minus;1)/2</sup> = ±1 (mod&nbsp;''p'')</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The equation </del>can <del style="font-weight: bold; text-decoration: none;">be tested rather quickly</del>, <del style="font-weight: bold; text-decoration: none;">which </del>can <del style="font-weight: bold; text-decoration: none;">be used for probabilistic [[prime testing|primality testing]]</del>. <del style="font-weight: bold; text-decoration: none;">These tests are twice </del>as <del style="font-weight: bold; text-decoration: none;">strong </del>as <del style="font-weight: bold; text-decoration: none;">tests based on Fermat's little theorem.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Every Euler pseudoprime is also a Fermat [[pseudoprime]]</del>. <del style="font-weight: bold; text-decoration: none;">It is not possible to produce </del>a <del style="font-weight: bold; text-decoration: none;">definite test </del>of <del style="font-weight: bold; text-decoration: none;">primality based on whether a [[number]] is an Euler pseudoprime because there exist ''absolute Euler pseudoprimes''</del>, <del style="font-weight: bold; text-decoration: none;">numbers </del>which are <del style="font-weight: bold; text-decoration: none;">Euler pseudoprimes to every base relatively prime to themselves</del>. The <del style="font-weight: bold; text-decoration: none;">absolute Euler pseudoprimes </del>are <del style="font-weight: bold; text-decoration: none;">a [[subset]] of the absolute Fermat pseudoprimes, or [[Carmichael number]]s, </del>and the <del style="font-weight: bold; text-decoration: none;">smallest absolute Euler pseudoprime </del>is <del style="font-weight: bold; text-decoration: none;">[[1729 (number)|1729]] = 7&times;13&times;19</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The slightly stronger condition </del>that</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">: </del><<del style="font-weight: bold; text-decoration: none;">math</del>> <del style="font-weight: bold; text-decoration: none;">a^{(n-1)/2} \equiv \left(\frac{a}{n}\right) \pmod n</del><<del style="font-weight: bold; text-decoration: none;">/math</del>></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">where ''n'' is an odd composite</del>, <del style="font-weight: bold; text-decoration: none;">the [[greatest common divisor]] of ''a'' and ''n'' equals 1</del>, <del style="font-weight: bold; text-decoration: none;">and (''</del>a<del style="font-weight: bold; text-decoration: none;">''/''n'') </del>is the <del style="font-weight: bold; text-decoration: none;">[[Jacobi symbol]], </del>is <del style="font-weight: bold; text-decoration: none;">the more common definition of an Euler pseudoprime</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">See, for example, page 115 of </del>the <del style="font-weight: bold; text-decoration: none;">book by Koblitz listed below, page 90 of the book by Riesel, or page 1003 of</del>.<del style="font-weight: bold; text-decoration: none;"><ref name="PSW">{{cite journal|coauthors = [[John L. Selfridge]], [[Samuel S. Wagstaff, Jr.]]|title=</del>The <del style="font-weight: bold; text-decoration: none;">pseudoprimes </del>to <del style="font-weight: bold; text-decoration: none;">25·10</del><<del style="font-weight: bold; text-decoration: none;">sup</del>><del style="font-weight: bold; text-decoration: none;">9</del><<del style="font-weight: bold; text-decoration: none;">/sup</del>><del style="font-weight: bold; text-decoration: none;">|journal=Mathematics of Computation|date=July 1980|volume=35|issue=151|pages=1003–1026|url=</del>http://www.<del style="font-weight: bold; text-decoration: none;">math</del>.<del style="font-weight: bold; text-decoration: none;">dartmouth.edu</del>/<del style="font-weight: bold; text-decoration: none;">~carlp/PDF/paper25.pdf|author = [[Carl Pomerance</del>]<del style="font-weight: bold; text-decoration: none;">]| doi=10.1090/S0025-5718-1980-0572872-7 }}</ref></del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">A discussion of numbers of this form can be found </del>at <del style="font-weight: bold; text-decoration: none;">[[Euler&ndash;Jacobi pseudoprime]].</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==See also==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* [[Probable prime]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==References==</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{reflist}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*M. Koblitz, "A Course in Number Theory and Cryptography", Springer</del>-<del style="font-weight: bold; text-decoration: none;">Verlag, 1987</del>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">*H. Riesel, "Prime numbers and computer methods of factorisation", Birkhäuser, Boston, Mass., 1985.</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{Classes of natural numbers}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Euler Pseudoprime}}</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Pseudoprimes]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>77.70.120.85https://en.formulasearchengine.com/index.php?title=Chain_(algebraic_topology)&diff=3133&oldid=preven>David Eppstein: wikilink formal linear combination2013-11-29T19:12:05Z<p>wikilink <a href="/wiki/Free_abelian_group" title="Free abelian group">formal linear combination</a></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:12, 29 November 2013</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><br><br></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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 </ins></div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">The days of running </del>to <del style="font-weight: bold; text-decoration: none;">answer </del>the <del style="font-weight: bold; text-decoration: none;">phone before it quits ringing is over</del>. <del style="font-weight: bold; text-decoration: none;">Yes the cordless phone has been around </del>for <del style="font-weight: bold; text-decoration: none;">years but </del>this <del style="font-weight: bold; text-decoration: none;">Sony cordless phone offers quite most regular cordless phone systems. Functions on this cordless phone 2.4 ghz are great</del>.<<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">Losing weight isn</del>'<del style="font-weight: bold; text-decoration: none;">t simple - this were</del>, <del style="font-weight: bold; text-decoration: none;">it wouldn</del>'<del style="font-weight: bold; text-decoration: none;">t </del>be <del style="font-weight: bold; text-decoration: none;">a new well studied topic. Points help you lose weight faster than the others. 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Lots different forms of spy machines are provided by we ePathChina</del>.<<del style="font-weight: bold; text-decoration: none;">br</del>><<del style="font-weight: bold; text-decoration: none;">br</del>><del style="font-weight: bold; text-decoration: none;">In case you have any queries about where by and tips on how to utilize [</del>http://www.<del style="font-weight: bold; text-decoration: none;">amj-uk</del>.<del style="font-weight: bold; text-decoration: none;">com</del>/-<del style="font-weight: bold; text-decoration: none;">IT</del>-<del style="font-weight: bold; text-decoration: none;">Support</del>-.<del style="font-weight: bold; text-decoration: none;">html London IT Support</del>], <del style="font-weight: bold; text-decoration: none;">you can e mail us from our own site</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">: <math>a^{(n-1)/2} \equiv \pm 1\pmod{n}</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(where ''mod'' refers </ins>to the <ins style="font-weight: bold; text-decoration: none;">[[modular arithmetic|modulo]] operation)</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The motivation </ins>for this <ins style="font-weight: bold; text-decoration: none;">definition is the fact that all [[prime number]]s ''p'' satisfy the above equation which can be deduced from [[Fermat's little theorem]]</ins>. <ins style="font-weight: bold; text-decoration: none;">Fermat's theorem asserts that if ''p'' is prime, and coprime to ''a'', then ''a''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">''p''&minus;1</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>> <ins style="font-weight: bold; text-decoration: none;">= 1 (mod ''p</ins>'<ins style="font-weight: bold; text-decoration: none;">'). Suppose that ''p''>2 is prime</ins>, <ins style="font-weight: bold; text-decoration: none;">then </ins>'<ins style="font-weight: bold; text-decoration: none;">'p'' can </ins>be <ins style="font-weight: bold; text-decoration: none;">expressed </ins>as <ins style="font-weight: bold; text-decoration: none;">2''q''&nbsp;+&nbsp;1 where ''q'' is an integer</ins>. <ins style="font-weight: bold; text-decoration: none;">Thus; ''a''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">(2''q''+1)&nbsp;&minus;&nbsp;1</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>> <ins style="font-weight: bold; text-decoration: none;">= 1 (mod&nbsp;''p'') which means that ''</ins>a<ins style="font-weight: bold; text-decoration: none;">''<sup>2''q''</sup>&nbsp;&minus;&nbsp;</ins>1 <ins style="font-weight: bold; text-decoration: none;">= 0 (mod ''p'</ins>'<ins style="font-weight: bold; text-decoration: none;">)</ins>. <ins style="font-weight: bold; text-decoration: none;">This can </ins>be <ins style="font-weight: bold; text-decoration: none;">factored as (''</ins>a<ins style="font-weight: bold; text-decoration: none;">''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">''q''</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>><ins style="font-weight: bold; text-decoration: none;">&nbsp;&minus;&nbsp;1)('</ins>'a''<ins style="font-weight: bold; text-decoration: none;"><sup>''q''</sup> + 1) = 0 (mod '</ins>'<ins style="font-weight: bold; text-decoration: none;">p</ins>''<ins style="font-weight: bold; text-decoration: none;">) which is equivalent to </ins>''a<ins style="font-weight: bold; text-decoration: none;">''</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">(''p''&minus;1)/2</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>> <ins style="font-weight: bold; text-decoration: none;">= ±1 (mod&nbsp;''p'').</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The equation can be tested rather quickly, which can be used </ins>for <ins style="font-weight: bold; text-decoration: none;">probabilistic [[prime testing|primality testing]]</ins>. <ins style="font-weight: bold; text-decoration: none;">These tests are twice as strong as tests based on Fermat</ins>'<ins style="font-weight: bold; text-decoration: none;">s little theorem</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Every Euler pseudoprime is also a Fermat [[pseudoprime]]</ins>. <ins style="font-weight: bold; text-decoration: none;">It is not possible </ins>to <ins style="font-weight: bold; text-decoration: none;">produce </ins>a <ins style="font-weight: bold; text-decoration: none;">definite test of primality based on whether a [[number]] is an Euler pseudoprime because there exist ''absolute Euler pseudoprimes'', numbers which </ins>are <ins style="font-weight: bold; text-decoration: none;">Euler pseudoprimes to every base relatively prime to themselves</ins>. The <ins style="font-weight: bold; text-decoration: none;">absolute Euler pseudoprimes are </ins>a <ins style="font-weight: bold; text-decoration: none;">[[subset]] of </ins>the <ins style="font-weight: bold; text-decoration: none;">absolute Fermat pseudoprimes, or [[Carmichael number]]s, </ins>and the <ins style="font-weight: bold; text-decoration: none;">smallest absolute Euler pseudoprime is [[1729 (number)|1729]] = 7&times;13&times;19</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The slightly stronger condition that</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">: </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>> <ins style="font-weight: bold; text-decoration: none;">a^{(n-1)/2} \equiv \left(\frac{a}{n}\right) \pmod n</ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where ''n'' </ins>is an <ins style="font-weight: bold; text-decoration: none;">odd composite, </ins>the <ins style="font-weight: bold; text-decoration: none;">[[greatest common divisor]] </ins>of <ins style="font-weight: bold; text-decoration: none;">''a'' and ''n'' equals 1, and (''a''/''n'') is </ins>the <ins style="font-weight: bold; text-decoration: none;">[[Jacobi symbol]]</ins>, <ins style="font-weight: bold; text-decoration: none;">is </ins>the <ins style="font-weight: bold; text-decoration: none;">more common definition </ins>of <ins style="font-weight: bold; text-decoration: none;">an Euler pseudoprime</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">See, for example</ins>, <ins style="font-weight: bold; text-decoration: none;">page 115 of </ins>the <ins style="font-weight: bold; text-decoration: none;">book by Koblitz listed below, page 90 </ins>of <ins style="font-weight: bold; text-decoration: none;">the book by Riesel</ins>, <ins style="font-weight: bold; text-decoration: none;">or page 1003 of.<ref name=</ins>"<ins style="font-weight: bold; text-decoration: none;">PSW</ins>"><ins style="font-weight: bold; text-decoration: none;">{{cite journal|coauthors = [[John L. Selfridge]]</ins>, <ins style="font-weight: bold; text-decoration: none;">[[Samuel S</ins>. <ins style="font-weight: bold; text-decoration: none;">Wagstaff</ins>, <ins style="font-weight: bold; text-decoration: none;">Jr</ins>.<ins style="font-weight: bold; text-decoration: none;">]]|title=The pseudoprimes to 25·10</ins><<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">9</ins><<ins style="font-weight: bold; text-decoration: none;">/sup</ins>><ins style="font-weight: bold; text-decoration: none;">|journal=Mathematics of Computation|date=July 1980|volume=35|issue=151|pages=1003–1026|url=</ins>http://www.<ins style="font-weight: bold; text-decoration: none;">math.dartmouth.edu/~carlp/PDF/paper25.pdf|author = [[Carl Pomerance]]| doi=10</ins>.<ins style="font-weight: bold; text-decoration: none;">1090</ins>/<ins style="font-weight: bold; text-decoration: none;">S0025</ins>-<ins style="font-weight: bold; text-decoration: none;">5718</ins>-<ins style="font-weight: bold; text-decoration: none;">1980</ins>-<ins style="font-weight: bold; text-decoration: none;">0572872-7 }}</ref></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A discussion of numbers of this form can be found at [[Euler&ndash;Jacobi pseudoprime]]</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==See also==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Probable prime</ins>]<ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==References==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{reflist}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*M. Koblitz, "A Course in Number Theory and Cryptography"</ins>, <ins style="font-weight: bold; text-decoration: none;">Springer-Verlag, 1987.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*H. Riesel, "Prime numbers and computer methods of factorisation", Birkhäuser, Boston, Mass., 1985</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Classes of natural numbers}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Euler Pseudoprime}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Pseudoprimes]]</ins></div></td></tr>
</table>en>David Eppsteinhttps://en.formulasearchengine.com/index.php?title=Chain_(algebraic_topology)&diff=226445&oldid=prev77.102.166.0: /* Boundary operator on chains */ removed surplus equals signs.2011-05-21T17:09:45Z<p><span dir="auto"><span class="autocomment">Boundary operator on chains: </span> removed surplus equals signs.</span></p>
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