Prime gaps - Revision history

https://michaelnielsen.org/polymath/index.php?action=history&feed=atom&title=Prime_gaps Prime gaps - Revision history 2026-04-13T15:53:14Z Revision history for this page on the wiki MediaWiki 1.42.3 https://michaelnielsen.org/polymath/index.php?title=Prime_gaps&diff=2351&oldid=prev Asdf: replace an external link with an interwiki link 2009-08-19T23:39:50Z

<p>replace an external link with an interwiki link</p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:39, 19 August 2009</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</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># R. C. Baker and G. Harman, “The difference between consecutive primes,” Proc. Lond. Math. Soc., series 3, 72 (1996) 261–280. MR 96k:11111</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># R. C. Baker and G. Harman, “The difference between consecutive primes,” Proc. Lond. Math. Soc., series 3, 72 (1996) 261–280. MR 96k:11111</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;"><div># K. Soundararajan, [http://arxiv.org/abs/math/0605696 Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim]</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># K. Soundararajan, [http://arxiv.org/abs/math/0605696 Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim]</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># [<del style="font-weight: bold; text-decoration: none;">http</del>:<del style="font-weight: bold; text-decoration: none;">//en.wikipedia.org/wiki/</del>Prime_gap The Wikipedia entry on prime gaps]</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;">[wikipedia</ins>:Prime_gap<ins style="font-weight: bold; text-decoration: none;">|</ins>The Wikipedia entry on prime gaps<ins style="font-weight: bold; text-decoration: none;">]</ins>]</div></td></tr> </table>

Asdf https://michaelnielsen.org/polymath/index.php?title=Prime_gaps&diff=2267&oldid=prev Teorth at 13:50, 9 August 2009 2009-08-09T13:50:11Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 06:50, 9 August 2009</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td> <td colspan="2" class="diff-lineno">Line 5:</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>A recent result of Goldston-Pintz-Yildirim shows that there exist infinitely many n for which the gap is as small as <math>o(\log p_n)</math> (in fact more precise bounds are known). But the set of small gaps established by this method is sparse.</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>A recent result of Goldston-Pintz-Yildirim shows that there exist infinitely many n for which the gap is as small as <math>o(\log p_n)</math> (in fact more precise bounds are known). But the set of small gaps established by this method is sparse.</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>[[Cramer's conjecture]] asserts that the prime gap never exceeds <math>O(\log^2 p_n)</math> in size. If so, this resolves the [[finding primes]] project positively. However, the best upper bound on the prime gap is <math>O( p_n^{1/2} \log p_n )</math> assuming the Riemann hypothesis, and <math>O( p_n^{0.<del style="font-weight: bold; text-decoration: none;">535</del>} )</math> otherwise (a result of Baker and Harman).</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>[[Cramer's conjecture]] asserts that the prime gap never exceeds <math>O(\log^2 p_n)</math> in size. If so, this resolves the [[finding primes]] project positively. However, the best upper bound on the prime gap is <math>O( p_n^{1/2} \log p_n )</math> assuming the Riemann hypothesis, and <math>O( p_n^{0.<ins style="font-weight: bold; text-decoration: none;">525</ins>} )</math> otherwise (a result of <ins style="font-weight: bold; text-decoration: none;">Baker, Harman, and Pintz; an earlier bound of <math>O(p_n^{0.535})</math> was obtained by </ins>Baker and Harman<ins style="font-weight: bold; text-decoration: none;">.</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>Rankin showed that the prime gap can be as large as <math>\log p_n \frac{\log \log p_n \log \log \log \log p_n}{(\log \log \log p_n)^3}</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>Rankin showed that the prime gap can be as large as <math>\log p_n \frac{\log \log p_n \log \log \log \log p_n}{(\log \log \log p_n)^3}</math>.</div></td></tr> </table>

Teorth https://michaelnielsen.org/polymath/index.php?title=Prime_gaps&diff=2252&oldid=prev Teorth at 17:59, 8 August 2009 2009-08-08T17:59:59Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:59, 8 August 2009</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</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># R. C. Baker and G. Harman, “The difference between consecutive primes,” Proc. Lond. Math. Soc., series 3, 72 (1996) 261–280. MR 96k:11111</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># R. C. Baker and G. Harman, “The difference between consecutive primes,” Proc. Lond. Math. Soc., series 3, 72 (1996) 261–280. MR 96k:11111</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;"><div># K. Soundararajan, [http://arxiv.org/abs/math/0605696 Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim]</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># K. Soundararajan, [http://arxiv.org/abs/math/0605696 Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim]</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;"># [http://en.wikipedia.org/wiki/Prime_gap The Wikipedia entry on prime gaps]</ins></div></td></tr> </table>

Teorth https://michaelnielsen.org/polymath/index.php?title=Prime_gaps&diff=2250&oldid=prev Teorth: New page: If <math>p_n</math> denotes the n^th prime, then <math>p_{n+1}-p_n</math> is the n^th prime gap. On average, the prime number theorem tells us that <math>p_{n+1}-p_n</math> has size <math... 2009-08-08T17:58:54Z

<p>New page: If <math>p_n</math> denotes the n^th prime, then <math>p_{n+1}-p_n</math> is the n^th prime gap. On average, the prime number theorem tells us that <math>p_{n+1}-p_n</math> has size <math...</p> <p><b>New page</b></p><div>If <math>p_n</math> denotes the n^th prime, then <math>p_{n+1}-p_n</math> is the n^th prime gap.<br /> <br /> On average, the prime number theorem tells us that <math>p_{n+1}-p_n</math> has size <math>O(\log p_n)</math>. <br /> <br /> A recent result of Goldston-Pintz-Yildirim shows that there exist infinitely many n for which the gap is as small as <math>o(\log p_n)</math> (in fact more precise bounds are known). But the set of small gaps established by this method is sparse.<br /> <br /> [[Cramer's conjecture]] asserts that the prime gap never exceeds <math>O(\log^2 p_n)</math> in size. If so, this resolves the [[finding primes]] project positively. However, the best upper bound on the prime gap is <math>O( p_n^{1/2} \log p_n )</math> assuming the Riemann hypothesis, and <math>O( p_n^{0.535} )</math> otherwise (a result of Baker and Harman).<br /> <br /> Rankin showed that the prime gap can be as large as <math>\log p_n \frac{\log \log p_n \log \log \log \log p_n}{(\log \log \log p_n)^3}</math>.<br /> <br /> # R. C. Baker and G. Harman, “The difference between consecutive primes,” Proc. Lond. Math. Soc., series 3, 72 (1996) 261–280. MR 96k:11111<br /> # K. Soundararajan, [http://arxiv.org/abs/math/0605696 Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim]</div>

Teorth