Predicting Maximal Gaps in Sets of Primes
Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>></mo> <mi>r</mi> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>...
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2019-05-01
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Online Access: | https://www.mdpi.com/2227-7390/7/5/400 |
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doaj-7fa146c204554937ba41a1995021ab71 |
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record_format |
Article |
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DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Alexei Kourbatov Marek Wolf |
spellingShingle |
Alexei Kourbatov Marek Wolf Predicting Maximal Gaps in Sets of Primes Mathematics Cramér conjecture Gumbel distribution prime gap prime <i>k</i>-tuple residue class Shanks conjecture totient |
author_facet |
Alexei Kourbatov Marek Wolf |
author_sort |
Alexei Kourbatov |
title |
Predicting Maximal Gaps in Sets of Primes |
title_short |
Predicting Maximal Gaps in Sets of Primes |
title_full |
Predicting Maximal Gaps in Sets of Primes |
title_fullStr |
Predicting Maximal Gaps in Sets of Primes |
title_full_unstemmed |
Predicting Maximal Gaps in Sets of Primes |
title_sort |
predicting maximal gaps in sets of primes |
publisher |
MDPI AG |
series |
Mathematics |
issn |
2227-7390 |
publishDate |
2019-05-01 |
description |
Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>></mo> <mi>r</mi> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> be coprime integers. Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> be an increasing sequence of primes <i>p</i> satisfying two conditions: (i) <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>≡</mo> <mi>r</mi> </mrow> </semantics> </math> </inline-formula> (mod <i>q</i>) <i>and</i> (ii) <i>p</i> starts a prime <i>k</i>-tuple with a given pattern <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi></semantics></math></inline-formula>. Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>π</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> be the number of primes in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> not exceeding <i>x</i>. We heuristically derive formulas predicting the growth trend of the maximal gap <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo movablelimits="true" form="prefix">max</mo> <mrow> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> between successive primes <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>,</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>∈</mo> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi></msub></mrow></semantics></math></inline-formula>. Extensive computations for primes up to <inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>14</mn> </msup> </semantics> </math> </inline-formula> show that a simple trend formula <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>π</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <msub> <mi>π</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>O</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> works well for maximal gaps between initial primes of <i>k</i>-tuples with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> (e.g., twin primes, prime triplets, etc.) in residue class <i>r</i> (mod <i>q</i>). For <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn></mrow></semantics></math></inline-formula>, however, a more sophisticated formula <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>π</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mo>·</mo> <mfenced separators="" open="(" close=")"> <mi>log</mi> <mfrac> <mrow> <msubsup> <mi>π</mi> <mrow> <mi>c</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mi>x</mi> </mfrac> <mo>+</mo> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> </semantics> </math> </inline-formula> gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes (<inline-formula><math display="inline"><semantics><mrow><mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn></mrow></semantics></math></inline-formula>). The distribution of appropriately rescaled maximal gaps <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramér’s conjecture. We also conjecture that the number of maximal gaps between primes in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> below <i>x</i> is <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>O</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mi>x</mi> <mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. |
topic |
Cramér conjecture Gumbel distribution prime gap prime <i>k</i>-tuple residue class Shanks conjecture totient |
url |
https://www.mdpi.com/2227-7390/7/5/400 |
work_keys_str_mv |
AT alexeikourbatov predictingmaximalgapsinsetsofprimes AT marekwolf predictingmaximalgapsinsetsofprimes |
_version_ |
1725061696063537152 |
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doaj-7fa146c204554937ba41a1995021ab712020-11-25T01:36:39ZengMDPI AGMathematics2227-73902019-05-017540010.3390/math7050400math7050400Predicting Maximal Gaps in Sets of PrimesAlexei Kourbatov0Marek Wolf1JavaScripter.net, 15127 NE 24th St., #578, Redmond, WA 98052, USAFaculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University, Wóycickiego 1/3, Bldg. 21, PL-01-938 Warsaw, PolandLet <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>></mo> <mi>r</mi> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> be coprime integers. Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> be an increasing sequence of primes <i>p</i> satisfying two conditions: (i) <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>≡</mo> <mi>r</mi> </mrow> </semantics> </math> </inline-formula> (mod <i>q</i>) <i>and</i> (ii) <i>p</i> starts a prime <i>k</i>-tuple with a given pattern <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">H</mi></semantics></math></inline-formula>. Let <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>π</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> be the number of primes in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> not exceeding <i>x</i>. We heuristically derive formulas predicting the growth trend of the maximal gap <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo movablelimits="true" form="prefix">max</mo> <mrow> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> between successive primes <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>,</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>∈</mo> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi></msub></mrow></semantics></math></inline-formula>. Extensive computations for primes up to <inline-formula> <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>14</mn> </msup> </semantics> </math> </inline-formula> show that a simple trend formula <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>π</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <msub> <mi>π</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>O</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> works well for maximal gaps between initial primes of <i>k</i>-tuples with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> (e.g., twin primes, prime triplets, etc.) in residue class <i>r</i> (mod <i>q</i>). For <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn></mrow></semantics></math></inline-formula>, however, a more sophisticated formula <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <mfrac> <mi>x</mi> <mrow> <msub> <mi>π</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> <mo>·</mo> <mfenced separators="" open="(" close=")"> <mi>log</mi> <mfrac> <mrow> <msubsup> <mi>π</mi> <mrow> <mi>c</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mi>x</mi> </mfrac> <mo>+</mo> <mi>O</mi> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> </semantics> </math> </inline-formula> gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes (<inline-formula><math display="inline"><semantics><mrow><mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn></mrow></semantics></math></inline-formula>). The distribution of appropriately rescaled maximal gaps <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramér’s conjecture. We also conjecture that the number of maximal gaps between primes in <inline-formula> <math display="inline"> <semantics> <msub> <mi mathvariant="double-struck">P</mi> <mi>c</mi> </msub> </semantics> </math> </inline-formula> below <i>x</i> is <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>O</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>log</mi> <mi>x</mi> <mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/7/5/400Cramér conjectureGumbel distributionprime gapprime <i>k</i>-tupleresidue classShanks conjecturetotient |