Ellipses and hyperbolas of decompositions of even numbers into pairs of prime numbers.

[Gennady Butov]

This is just an attempt to associate sums or differences of prime numbers with points lying on an ellipse or hyperbola.
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.

1. The ellipse equation can be written in the following form:  |p(k)| + |p(t)| = 2n

2. The hyperbola equation can be written in the following form:  ||p(k)| - |p(t)|| = 2n                         
 
where p(k) and p(t) are prime numbers (p(1) = 2, p(2) = 3, p(3) = 5, p(4) = 7,...),
k and t are indices of prime numbers,
2n is a given even number,
k, t, n ∈ N.

If we construct ellipses and hyperbolas based on the above, we get the following:

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.

2. There is only 1 non-intersecting hyperbola (for 2n=2) and 1 non-intersecting vertical line. The remaining hyperbolas have intersection points.

Will there be any new thoughts, ideas about this?

ellipses_hyperbolas_decompositions.pdf

[Trurl]

I don’t understand. I have been on the opposite end of this in my simple yet interesting post. It just goes to prove my theory that a pattern of Primes will never happen. Not because there isn’t a pattern but because the thoughts that lead to a pattern cannot be communicated from one person to another.

 

For my post it came down to if I could factor a large SemiPrime or not. I couldn’t do it efficiently. But could reflect my programming skills and not my equations.

 

Just as I started, your write up is too complex. It would take months to sort it out and it could be wrong. I have the same troubles when explaining. Give me an example where I can plug and chug.

 

I am always interested in Primes. Why don’t you elaborate on the diagram of the ellipse.