July 1, 2023 Preprint Open Access

Gennady Butov

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  <identifier identifierType="DOI">10.15161/oar.it/77110</identifier>
  <creators>
    <creator>
      <creatorName>Gennady Butov</creatorName>
    </creator>
  </creators>
  <titles>
    <title>Ellipses and hyperbolas of decomposition of even numbers into pairs of prime numbers</title>
  </titles>
  <publisher>INFN Open Access Repository</publisher>
  <publicationYear>2023</publicationYear>
  <subjects>
    <subject>prime, primes, number, numbers, even, theory, ellipse, ellipses, hyperbole, hyperboles, decomposition</subject>
    <subject subjectScheme="pmid">11</subject>
  </subjects>
  <dates>
    <date dateType="Issued">2023-07-01</date>
  </dates>
  <language>en</language>
  <resourceType resourceTypeGeneral="Text">Preprint</resourceType>
  <alternateIdentifiers>
    <alternateIdentifier alternateIdentifierType="url">https://www.openaccessrepository.it/record/77110</alternateIdentifier>
  </alternateIdentifiers>
  <relatedIdentifiers>
    <relatedIdentifier relatedIdentifierType="DOI" relationType="IsVersionOf">10.15161/oar.it/77082</relatedIdentifier>
  </relatedIdentifiers>
  <rightsList>
    <rights rightsURI="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution 4.0</rights>
    <rights rightsURI="info:eu-repo/semantics/openAccess">Open Access</rights>
  </rightsList>
  <descriptions>
    <description descriptionType="Abstract">&lt;p&gt;This is just an attempt to associate sums or differences of prime numbers with points lying on an ellipse or hyperbola.&lt;br&gt;
Certain pairs of prime numbers can be represented as radius-distances from the focuses to points lying either on the ellipse or on the hyperbola.&lt;br&gt;
The ellipse equation can be written in the following form: |p(k)| + |p(t)| = 2n.&lt;br&gt;
The hyperbola equation can be written in the following form: ||p(k)| - |p(t)|| = 2n.&lt;br&gt;
Here p(k) and p(t) are prime numbers (p(1) = 2, p(2) = 3, p(3) = 5, p(4) = 7,...),&lt;br&gt;
k and t are indices of prime numbers,&lt;br&gt;
2n is a given even number,&lt;br&gt;
k, t, n &amp;isin; N.&lt;br&gt;
If we construct ellipses and hyperbolas based on the above, we get the following:&lt;br&gt;
1) there are only 5 non-intersecting curves (for 2n=4; 2n=6; 2n=8; 2n=10; 2n=16). The remaining ellipses have intersection points.&lt;br&gt;
2) there is only 1 non-intersecting hyperbola (for 2n=2) and 1 non-intersecting vertical line. The remaining hyperbolas have intersection points.&lt;br&gt;
Will there be any new thoughts, ideas about this?&lt;/p&gt;</description>
  </descriptions>
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