July 1, 2023 Preprint Open Access

Gennady Butov

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  "@id": "https://doi.org/10.15161/oar.it/77110", 
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      "@id": "http://www.ncbi.nlm.nih.gov/pubmed/11", 
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  "contributor": [
    {
      "@type": "Person", 
      "name": "Gennady Butov"
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  "creator": [
    {
      "@type": "Person", 
      "name": "Gennady Butov"
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  "datePublished": "2023-07-01", 
  "description": "<p>This is just an attempt to associate sums or differences of prime numbers with points lying on an ellipse or hyperbola.<br>\nCertain pairs of prime numbers can be represented as radius-distances from the focuses to points lying either on the ellipse or on the hyperbola.<br>\nThe ellipse equation can be written in the following form: |p(k)| + |p(t)| = 2n.<br>\nThe hyperbola equation can be written in the following form: ||p(k)| - |p(t)|| = 2n.<br>\nHere p(k) and p(t) are prime numbers (p(1) = 2, p(2) = 3, p(3) = 5, p(4) = 7,...),<br>\nk and t are indices of prime numbers,<br>\n2n is a given even number,<br>\nk, t, n &isin; N.<br>\nIf we construct ellipses and hyperbolas based on the above, we get the following:<br>\n1) there are only 5 non-intersecting curves (for 2n=4; 2n=6; 2n=8; 2n=10; 2n=16). The remaining ellipses have intersection points.<br>\n2) there is only 1 non-intersecting hyperbola (for 2n=2) and 1 non-intersecting vertical line. The remaining hyperbolas have intersection points.<br>\nWill there be any new thoughts, ideas about this?</p>", 
  "headline": "Ellipses and hyperbolas of decomposition of even numbers into pairs of prime numbers", 
  "identifier": "https://doi.org/10.15161/oar.it/77110", 
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  "keywords": [
    "prime, primes, number, numbers, even, theory, ellipse, ellipses, hyperbole, hyperboles, decomposition"
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  "license": "https://creativecommons.org/licenses/by/4.0/", 
  "name": "Ellipses and hyperbolas of decomposition of even numbers into pairs of prime numbers", 
  "url": "https://www.openaccessrepository.it/record/77110"
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