https://number.subwiki.org/w/index.php?action=history&feed=atom&title=Bitwin_chain
Bitwin chain - Revision history
2026-05-09T23:38:51Z
Revision history for this page on the wiki
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https://number.subwiki.org/w/index.php?title=Bitwin_chain&diff=510&oldid=prev
Vipul at 02:20, 2 May 2010
2010-05-02T02:20:37Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:20, 2 May 2010</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></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>A '''bitwin chain''' of length <math>k</math> is defined as a collection of natural numbers:</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>A '''bitwin chain''' of length <math>k <ins style="font-weight: bold; text-decoration: none;">+ 1</ins></math> is defined as a collection of natural numbers:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>(n-1,n+1,2n-1,2n+1, \dots 2^k\cdot n - 1, 2^k \cdot n + 1)</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>(n-1,n+1,2n-1,2n+1, \dots 2^k\cdot n - 1, 2^k \cdot n + 1)</math></div></td></tr>
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Vipul
https://number.subwiki.org/w/index.php?title=Bitwin_chain&diff=509&oldid=prev
Vipul at 02:17, 2 May 2010
2010-05-02T02:17:39Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:17, 2 May 2010</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>such that all the numbers in the chain are prime.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>such that all the numbers in the chain are prime.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></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>Note that the numbers <math>n-1, 2n-1, \cdots 2^kn - 1</math> forms a [[Cunningham chain of the first kind]] of length <math>k + 1</math>, while <math>n+1, 2n + 1, \dots, 2^kn + 1</math> forms a [[<del style="font-weight: bold; text-decoration: none;">Cunninghan </del>chain of the second kind]]. Each of the pairs <math>2^in - 1, 2^in+ 1</math> is a pair of [[twin primes]].</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>Note that the numbers <math>n-1, 2n-1, \cdots 2^kn - 1</math> forms a [[<ins style="font-weight: bold; text-decoration: none;">defining ingredient::</ins>Cunningham chain of the first kind]] of length <math>k + 1</math>, while <math>n+1, 2n + 1, \dots, 2^kn + 1</math> forms a [[<ins style="font-weight: bold; text-decoration: none;">defining ingredient::Cunningham </ins>chain of the second kind]]. Each of the pairs <math>2^in - 1, 2^in+ 1</math> is a pair of [[<ins style="font-weight: bold; text-decoration: none;">defining ingredient::</ins>twin primes<ins style="font-weight: bold; text-decoration: none;">]]. Each of the primes <math>2^in - 1</math> for <math>0 \le i \le k - 1</math> is a [[defining ingredient::Sophie Germain prime]] and each of the primes <math>2^in - 1</math> for <math>1 \le i \le k</math> is a [[defining ingredient::safe prime</ins>]].</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Relation with other properties==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Relation with other properties==</div></td></tr>
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Vipul
https://number.subwiki.org/w/index.php?title=Bitwin_chain&diff=100&oldid=prev
Vipul: Created page with '==Definition== A '''bitwin chain''' of length <math>k</math> is defined as a collection of natural numbers: <math>(n-1,n+1,2n-1,2n+1, \dots 2^k\cdot n - 1, 2^k \cdot n + 1)</ma...'
2009-04-20T18:30:53Z
<p>Created page with '==Definition== A '''bitwin chain''' of length <math>k</math> is defined as a collection of natural numbers: <math>(n-1,n+1,2n-1,2n+1, \dots 2^k\cdot n - 1, 2^k \cdot n + 1)</ma...'</p>
<p><b>New page</b></p><div>==Definition==<br />
<br />
A '''bitwin chain''' of length <math>k</math> is defined as a collection of natural numbers:<br />
<br />
<math>(n-1,n+1,2n-1,2n+1, \dots 2^k\cdot n - 1, 2^k \cdot n + 1)</math><br />
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such that all the numbers in the chain are prime.<br />
<br />
Note that the numbers <math>n-1, 2n-1, \cdots 2^kn - 1</math> forms a [[Cunningham chain of the first kind]] of length <math>k + 1</math>, while <math>n+1, 2n + 1, \dots, 2^kn + 1</math> forms a [[Cunninghan chain of the second kind]]. Each of the pairs <math>2^in - 1, 2^in+ 1</math> is a pair of [[twin primes]].<br />
<br />
==Relation with other properties==<br />
<br />
===Related chains===<br />
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* [[Cunningham chain of the first kind]]<br />
* [[Cunningham chain of the second kind]]<br />
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===Related properties of primes/pairs of primes===<br />
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* [[Twin primes]]<br />
* [[Sophie Germain prime]] is a prime <math>p</math> such that <math>2p + 1</math> is also prime.<br />
* [[Safe prime]] is a prime <math>p</math> such that <math>(p-1)/2</math> is also prime.</div>
Vipul