July 1, 2023 Preprint Open Access

Gennady Butov

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{
  "DOI": "10.15161/oar.it/77110", 
  "abstract": "<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>", 
  "author": [
    {
      "family": "Gennady Butov"
    }
  ], 
  "id": "77110", 
  "issued": {
    "date-parts": [
      [
        2023, 
        7, 
        1
      ]
    ]
  }, 
  "language": "eng", 
  "title": "Ellipses and hyperbolas of decomposition of even numbers into pairs of prime numbers", 
  "type": "article"
}