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1. I specifically began to consider ellipses and hyperbolas generated by sets of pairs of prime numbers.
I really wanted neighboring ellipses/hyperbolas to intersect in pairs - it would be nice and simple.
But the prime numbers again presented an unexpected surprise.
I noticed another peculiarity - not all neighboring (closest to each other) ellipses/hyperbols intersect each other.
2. The fact is that I didn't even prove anything. I have only put forward 2 hypotheses.
That is, my whole article is just a statement of the problem, not its solution.
The calculations that I have given are most similar to a computational experiment that shows a limited number of results does not cover the entire infinite set.
Someday someone will prove these two hypotheses.
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.
Equations of the ellipse: |p(k)| + |p(t)| = 2*n
Equations of the hyperbola: ||p(k)| - |p(t)|| = 2*n
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,
2*n is a given even number,
k, t, n ∈ N.
If we construct ellipses and hyperbolas based on the above, we get the following:
1. Hypothesis of intersecting decomposition ellipses: 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. Hypothesis of intersecting decomposition hyperbolas: there is only 1 non-intersecting hyperbola (for 2n=2) and 1 non-intersecting vertical line. The remaining hyperbolas have intersection points.
hi Bob, here is a link to an article on this topic https://drive.google.com/file/d/1CBQyNyKmOlU9k6oP_Wp1wx2t3TXBbnKi/view
1. The sum of two prime numbers is represented as an ellipse equation. A hypothesis is put forward about the infinity of intersecting ellipses given in the form of equations with prime numbers. The conditions for the intersection of ellipses are determined when decomposing even numbers into two primes.
2. The differences of two prime numbers are represented as an equation of a hyperbola. A hypothesis is put forward about the infinity of intersecting hyperbolas, which are given in the form of equations with prime numbers. The conditions for the intersection of hyperbolas are determined when decomposing even numbers into two primes.
Will there be any new thoughts, ideas about this?
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