Primes distribution

I got the distribution from the diagram by prolonging the initial $1+$ and $1-$ curves on $Y$, getting the numbers that appeared on the converging points and bolding the prime ones. Then I compared those that appeared on $Y+$ and $Y-$.

Primes distribution detail

I found a similar diagram representing an example of Galois corresponde (Ref. "A Book of Abstract Algebra", Charles C. Pinter):

Galois Corrrespondence diagram

Prime numbers correspondence

On the other hand, adding also to $Y+$ and $Y-$ the numbers that converge when prolonging the curves from the point $9$ on $Y$ (I didn't draw thos prolongations on the diagram), the resultant distribution is this another one:

Full primes distribution

It also can be presented in this way:

Full primes distribution 2

  • 1
    $\begingroup$ I think you are going to explain this a lot more. What is Y+, Y-, 1+, 1-? What do you mean by converging points? You should be to explain your question without reference to the diagram. But, +1 because the diagram is somehow intriguing. $\endgroup$ – Jair Taylor May 25 '18 at 20:20
  • $\begingroup$ Hi Jair. i started the diagram from the + curve that starts from the zero point to the point $Y1$, and its conjugate - curve. Then I prolonged those curves in all the possible ways. When the opposite curves converge on Y I get the number of the converging point. $\endgroup$ – user6562 May 25 '18 at 23:03

I'm almost sure it doesn't have a name(other than what I describe below) since it isn't too interesting.

The curve just places different numbers$\bmod 8$ on $8$ rays from the origin. So, looking at your first distribution, you simply are creating

$$Y^+ = \{n \in \mathbb{N} \mid n \equiv 1,3 \pmod{8} \}$$ $$Y^- = \{n \in \mathbb{N} \mid n \equiv 5,7 \pmod{8} \}$$

It is somewhat interesting that pretty much every prime is on $Y^{+}$ or $Y^{-}$. To be precise, $\mathbb{P} \setminus \{2\} \subset Y^+ \cup Y^-$. But this is obvious when you analyze it, since every prime $p$ except $2$ is odd, so

$$p \not\equiv 0, 2, 4, 6 \pmod{8}$$

And indeed, you forgot to put $2$ in your prime distrubtion.

  • $\begingroup$ Thank you Jeffery, I could've set them on X or Z starting from other point, but what's in interesting to me is the way they appear on the positive or negative sides following the curves. The spacial distribution of the serie 7 to 83 describes a kind geometrical open "ring" with antisymmetric elements, (7,83), (11, 79), (19,71), (23, 67), (31,59), (43,47). The addition of each couple is = 90. The total primes of this serie are 12 and there are 6 antisymmetric couples. It reminds me in some way of the structure of the bencene molecule. 2 was not added because here it converged at Z. $\endgroup$ – user6562 May 25 '18 at 22:57

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