July 1, 2023 Preprint Open Access

Gennady Butov

MARC21 XML Export

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    <subfield code="a">&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;</subfield>
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    <subfield code="a">Ellipses and hyperbolas of decomposition of even numbers into pairs of prime numbers</subfield>
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