Prime Identity

Here we would like to compile the way we take on getting the arithmetic expresion of an individual unit expression (identity) such as a taxicab number below.

These three (3) number are twin primes. We called the pairs as True Prime Pairs. Our scenario is mapping the distribution out of these pairs by taking the symmetrical behaviour of 36 as the smallest power (greater than 1) which is not a prime power.

$True Prime Pairs:
 (5,7), (11,13), (17,19)
 
 layer|  i  |   f
 -----+-----+---------
      |  1  | 5
   1  +-----+
      |  2  | 7
 -----+-----+---  } 36 » 6®
      |  3  | 11
   2  +-----+
      |  4  | 13
 -----+-----+---------
      |  5  | 17
   3  +-----+     } 36 » 6®
      |  6  | 19
 -----+-----+---------

Thus in short this is all about the method that we called as the 19 vs 18 Scenario of mapping the quantum way within a huge of primes objects (5 to 19) by lexering (11) the ungrammared feed (7) and parsering (13) across syntax (17).

Φ(1,2,3) = Φ(6,12,18) = Φ(13,37,61)

$True Prime Pairs:
(5,7), (11,13), (17,19)
 
layer | node | sub |  i  |  f
------+------+-----+----------
      |      |     |  1  | 
      |      |  1  +-----+          
      |  1   |     |  2  | (5)
      |      |-----+-----+
      |      |     |  3  |
  1   +------+  2  +-----+----
      |      |     |  4  |
      |      +-----+-----+
      |  2   |     |  5  | (7)
      |      |  3  +-----+
      |      |     |  6  |
------+------+-----+-----+------      } (36)
      |      |     |  7  |
      |      |  4  +-----+
      |  3   |     |  8  | (11)
      |      +-----+-----+
      |      |     |  9  |
  2   +------|  5  +-----+-----
      |      |     |  10 |
      |      |-----+-----+
      |  4   |     |  11 | (13)
      |      |  6  +-----+
      |      |     |  12 |
------+------+-----+-----+------------------
      |      |     |  13 |
      |      |  7  +-----+
      |  5   |     |  14 | (17)
      |      |-----+-----+
      |      |     |  15 |
  3   +------+  8  +-----+-----       } (36)
      |      |     |  16 |
      |      |-----+-----+
      |  6   |     |  17 | (19)
      |      |  9  +-----+
      |      |     |  18 |
------|------|-----+-----+------

The main background is that, as you may also aware, the prime number theorem describes the asymptotic distribution of prime numbers which is still a major problem in mathematic.

Zeta Function

Instead of getting a proved formula we came to a unique expression called zeta function. This expression first appeared in a paper in 1737 entitled Variae observationes circa series infinitas.

This issue is actually come from Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered by many mathematicians to be the most important of unsolved problems in pure mathematics.

Meanwhile obtaining the non complex numbers it is easier to look at a graph like the one below which shows Li(x) (blue), R(x) (black), π(x) (red) and x/ln x (green); and then proclaim "R(x) is the best estimate of π(x)." Indeed it is for that range, but as we mentioned above, Li(x)-π(x) changes sign infinitely often, and near where it does, Li(x) would be the best value.

And we can see in the same way that the function Li(x)-(1/2)Li(x1/2) is ‘on the average' a better approximation than Li(x) to π(x); but no importance can be attached to the latter terms in Riemann's formula even by repeated averaging. The problem is that the contributions from the non-trivial zeros at times swamps that of any but the main terms in these expansions.

Moreover in it was verified numerically, in a rigorous way using interval arithmetic, that The Riemann hypothesis is true up to 3 · 10^12. That is, all zeroes β+iγ of the Riemann zeta-function with 0<γ≤3⋅1012 have β=1/2.

This Euler formula represents the distribution of a group of numbers that are positioned at regular intervals on a straight line to each other. Riemann later extended the definition of zeta(s) to all complex numbers (except the simple pole at s=1 with residue one). Euler's product still holds if the real part of s is greater than one. Riemann derived the functional equation of zeta function.

If both of the above statements are true then mathematically this Riemann Hypothesis is proven to be incorrect because it only applies to certain cases or limitations. So first of all the basis of the Riemann Hypothesis has to be considered.

Sehr leider Herr Riemann. Bis jetzt Leute können den Fall immer noch nicht lösen.

On the other hand, the possibility of obtaining the function of the distribution of prime numbers shall go backwards since it needs significant studies to be traced. Or may be start again from the Euleur Function.

Euler's identity

Freeman Dyson discovered an intriguing connection between quantum physics and Montgomery's pair correlation conjecture about the zeros of the zeta function which dealts with the distribution of primes.

The path plan assume that a symmetric distribution of prime numbers with equal axial lengths from a middle zero axis = 15 is able to determine the distribution of primes in a given number space. This assumption finally bring us to the equation of Euler's identity.

The finiteness position of Euler's identity by the said MEC30 opens up the possibility of accurately representing the self-similarity based on the distribution of True Prime Pairs so that all number would belongs together with their own identitities.

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See that there are multiple repetition from addition to multiplication which may lead up to the concept of 11th-dimension. This path is being applied as you can find on the left sidebar. (Please change the view to desktop mode if you are on mobile browser).

Nothing is going to be easly about the nature of prime numbers but they demonstrably congruent to something organized. Let's discuss starting with the addition zones.