Prime Unit Identity

The Prime Hexagon is a mathematical structure developed by mathematician Tad Gallion. A Prime Hexagon is formed when integers are sequentially added to a field of tessellating equilateral triangles, where the path of the integers is changed whenever a prime number is encountered (GitHub: kaustubhcs/prime-hexagon).

It is a taxicab number, and is variously known as Ramanujan’s number and the Ramanujan-Hardy number, after an anecdote of the British mathematician GH Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital (Wikipedia).

The number 36 is a composite. Here are some of the other points taken from (Prime Curios!).

• The exact number of ways to partition the integer 36 is prime.
• The smallest number which is the sum of two distinct odd primes in four ways (36 = 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19). [McCranie]
• The smallest square that is the sum of a twin prime pair {17, 19}. [Trotter]
• The smallest number expressible as the sum of consecutive primes in two ways (5 + 7 + 11 + 13 and 17 + 19). [De Geest]
• 5+7+11+13 is the smallest square number expressible as the sum of four consecutive primes which are also two couples of prime twins! [Herault]

Prime numbers are numbers that have only 2 factors: 1 and themselves.

• For example, the first 5 prime numbers are 2, 3, 5, 7, and 11. By contrast, numbers with more than 2 factors are call composite numbers.
• 1 is not a prime number because it can not be divided by any other integer except for 1 and itself. The only factor of 1 is 1.
• On the other hand, 1 is also not a composite number because it can not be divided by any other integer except for 1 and itself.

In conclusion, the number 1 is neither prime nor composite.

Similarly, I have a six colored dice in the form of the hexagon. If I take a known, logical sequence of numbers, say 10, 100, 1000, 10000, and look at their spins in the hexagon, the resulting colors associated with each number should appear random – unless the sequence I’m investigating is linked to the nature of the prime numbers.

In addition to the trivial roots, there also exist complex roots for real t. We find that the he first ten (10) non-trivial roots of the Riemann zeta function is occured when the values of t below 50. A plot of the values of ζ(1/2 + it) for t ranging from –50 to +50 is shown below. The roots occur each time the locus passes through the origin. (mathpages).

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.

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 Mathematical Elementary Cell 30 (MEC30) standard unites the mathematical and physical results of 1972 by the mathematician Hugh Montgomery and the physicist Freeman Dyson and thus reproduces energy distribution in systems as a path plan more accurately than a measurement. (Google Patent DE102011101032A9)

Euler’s identity is considered to be an exemplar of deep mathematical beauty as it shows a profound connection between the most fundamental numbers. Three (3) of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation (Wikipedia).

You likely noticed I began with 2 rather than 1 or 0 when I first constructed the hexagon. Why? Because they do not fit inside — they stick off the hexagon like a tail. Perhaps that’s where they belong. However, if one makes a significant and interesting assumption, then 1 and 0 fall in their logical locations – in the 1 and 0 cells, respectively. _(HexSpin)

In this one system, represented as an icon, we can see the distribution profile of the prime numbersas well as their products via a chessboard-like model in Fig. 4. This fundamental chewing

• We show the connection in the MEC 30 mathematically and precisely in the table Fig. 13. The organization of this table is based on the well-known idea of Christian Goldbach.
• That every even number from the should be the sum of two prime numbers. From now on we call all pairs of prime numbers without “1”, 2, 3, 5 Goldbach couples.

The MEC 30 transforms this idea from Christian Goldbach into the structure of a numerical double strand, into an opposite link of the MEC 30 scale. (MEC 30 - pdf)

The SM was basically developed in 1970-s. It describes the electromagnetic, weak and strong fundamental interactions.

• At ordinary energies (a few eV or less), the forces differ greatly. However, at energies available in accelerators, the weak nuclear and electromagnetic (EM) forces become unified. Unfortunately, the energies at which the strong nuclear and electroweak forces become the same are unreachable.
• The relative strengths of the four basic forces vary with distance, and, hence, energy is needed to probe small distances.
• The (3) layers represents generation in the particle objects of flavor that counts six (6) flavours of quarks and six (6) flavours of leptons.
• The newly discovered Higgs Boson interacts with all the Quarks and the first group of Leptons (electron, muon and tau) providing them with their mass. The neutrinos which are the other Leptons originally were thought to have zero mass, but recent discoveries argue that this is not the case.

The Weak bosons interact with both Leptons and Quarks, these are responsible for the Weak nuclear forces. The exchange of photon is responsible for the Electromagnetic Force.

In this one system, reproduced as an icon, we can show the distribution profile of the primes as well as their products over a checkerboard-like model in the 4.

• We show this fundamental causal relationship in the MEC 30 mathematically accurate in the table 13 , The organization of this table is based on the well-known idea of Christian Goldbach. That every even number should consist of the sum of two primes.
• All pairs of prime numbers without “1”, 2, 3, 5, we call henceforth Goldbach pairs. The MEC 30 transforms this idea of Christian Goldbach into the structure of a numerical double-strand, into an opposing member of the MEC 30 scale.
• We call this double strand a convolution, which results in an opposite arrangement. It represents the natural vibration, thus also the redundant vibrations in the energy transfer. In the 6 For example, in the graph, the even number 60 is folded. At folding of the even number 60 6 result in 8 prime pairs.
• In this case, among the 8 pairs of prime pairs there are only 6 Goldbach pairs. 2 prime positions in the prime position pairs carry products of the factors “1 × 1” and 7 × 7. Thus, 2 prime pairs do not fulfill the requirements of the Goldbach pairs. In general, any even number larger than 30 can be represented graphically within a cycle (MEC 30) as a specific cyclic convolution. This characteristic convolution of the even numbers is a fundamental test element in the numerical table. The result Even the even numbers to infinity occupy a fixed position within the 30s system MEC 30. The even numbers thus have 15 positions: 30/2 = 15 even positions of the MEC 30.
• There are therefore only 15 even positions for all even numbers to infinity. Every even number has a specific convolution due to its position in the 30s system. First, we have to determine the positions of the even numbers in the 30s system to make them one in the following graph 7 attributable to the 15 specific folds.

There are 8 different types of tiny particles, or ‘states’, that we can find in a special kind of space that has 6 dimensions and involves both real and imaginary numbers. These particles include:

• The Higgs field, which doesn’t spin and is represented by 0.
• Fermions, which are particles like electrons, having a spin of plus or minus a half.
• Bosons, like photons, which have a spin of plus or minus 1.
• Anti-fermions, which are like fermions but have a spin of plus or minus two-thirds.
• The graviton, believed to be responsible for gravity, with a spin of 2.

In a diagram at the top left, this 6-dimensional space is shown to be curved. In another diagram at the bottom right, we see two waves that are perpendicular to each other, representing the motion of a particle in a ‘Dirac harmonic oscillator’ – a concept in quantum mechanics. (Physics In History)

By our project the spin will be generated from 18’s on the gist so it will cover five (5) unique functions that behave as one (1) central plus four (4) zones.

• This scheme will be implemented to all of the 168 repositories as bilateral way (in-out) depend on their postion on the system. So along with the gist it self then there shall be 1 + 168 = 169 units of 1685 root functions. Each of functions has their own model. Those models will be organized per IREE’s plan below:
• IREE (Intermediate Representation Execution Environment) is an MLIR-based end-to-end compiler and runtime that lowers Machine Learning (ML) models to a unified IR that scales up to meet the needs of the datacenter and down to satisfy the constraints and special considerations of mobile and edge deployments.

From what we learned above about segregating spin candidates, we can demonstrate that they compile additively in perfect progression, completing an infinite sequence of circles (multiples of 30 and 360) which is inline with the behaviour of MEC30.

A thirt, in case you’re wondering, is a useful unit of measure when discussing intervals in natural numbers not divisible by 2, 3 or 5.

• Another fascinating feature of this array is that any even number of–not necessarily contiguous–factors drawn from any one of the 32 angles in this modulo 120 configuration distribute products to 1(mod 120) or 49 (mod 120), along with the squares.
• We see from the graphic above that the digital roots of the Fibonacci numbers indexed to our domain (Numbers ≌ to {1,7,11,13,17,19,23,29} modulo 30) repeat palindromically every 32 digits (or 4 thirts) consisting of 16 pairs of bilateral 9 sums.
• The digital root sequence of our domain, on the other hand, repeats every 24 digits (or 3 thirts) and possesses 12 pairs of bilateral 9 sums. The entire Prime Root sequence end-to-end covering 360° has 48 pairs of bilateral 9 sums.
• And finally, the Prime Root elements themselves within the Cirque, consisting of 96 elements, has 48 pairs of bilateral sums totaling 360. Essentially, the prime number highway consists of infinitely telescoping circles …
• Also note, the digital roots of the Prime Root Set as well as the digital roots of Fibonnaci numbers and Lucas numbers (the latter not shown above) indexed to it all sum to 432 (48x9) in 360° cycles.
• The sequence involving Fibonacci digital roots repeats every 120°, and has been documented by the author on the On-Line Encyclopedia of Integer Sequences: Digital root of Fibonacci numbers indexed by natural numbers not divisible by 2, 3 or 5 (A227896).
• The four faces of our pyramid additively cascade 32 four-times triangular numbers (Note that 4 x 32 = 128 = the perimeter of the square base which has an area of 32^2 = 1024 = 2^10).
• These include Fibo1-3 equivalent 112 (rooted in T7 = 28; 28 x 4 = 112), which creates a pyramidion or capstone in our model, and 2112 (rooted in T32 = 528; 528 x 4 = 2112), which is the index number of the 1000th prime within our domain, and equals the total number of ‘elements’ used to construct the pyramid.

If we take the Modulo 30 Prime Spiral Sieve and expand it to Modulo 360, we see that there are 12 thirts in one complete circle, or ‘cirque’ as we’ve dubbed it. Each thirt consists of 8 elements. (PrimesDemystified)

There is a fascinating connection between prime numbers and the Golden ratio.

• The Golden ratio is an irrational number, which means that it cannot be expressed as a ratio of two integers. However, it can be approximated by dividing consecutive Fibonacci numbers.
• Additionally, it has been observed that the frequency of prime numbers in certain sequences related to the Golden ratio (such as the continued fraction expansion of the Golden ratio) appears to be higher than in other sequences.
• Interestingly, the Fibonacci sequence is closely related to prime numbers, as any two consecutive Fibonacci numbers are always coprime.

However, the exact nature of the relationship between primes and the Golden ratio is still an active area of research.

Simply stated, the Golden Ratio establishes that the small is to the large as the large is to the whole. This is usually applied to proportions between segments.

• In the special case of a unit segment, the Golden Ratio provides the only way to divide unity in two parts that are in a geometric progression:
• The side of a pentagon-pentagram can clearly be seen as in relation to its diagonal as 1: (√5 +1)/2 or 1:φ, the Golden Section:
• When you draw all the diagonals in the pentagon you end up with the pentagram. The pentagram shows that the Golden Gnomon, and therefore Golden Ratio, are iteratively contained inside the pentagon:
• There are set of sequence known as Fibonacci retracement. For unknown reasons, these Fibonacci ratios seem to play a role in the stock market, just as they do in nature. The Fibonacci retracement levels are 0.236, 0.382, 0.618, and 0.786.
  • The key Fibonacci ratio of 61.8% is found by dividing one number in the series by the number that follows it. For example, 21 divided by 34 equals 0.6176, and 55 divided by 89 equals about 0.61798.
  • The 38.2% ratio is discovered by dividing a number in the series by the number located two spots to the right. For instance, 55 divided by 144 equals approximately 0.38194.
  • The 23.6% ratio is found by dividing one number in the series by the number that is three places to the right. For example, 8 divided by 34 equals about 0.23529.
  • The 78.6% level is given by the square root of 61.8%
• While not officially a Fibonacci ratio, 0.5 is also commonly referenced (50% is derived not from the Fibonacci sequence but rather from the idea that on average stocks retrace half their earlier movements).

This study cascade culminating in the Fibonacci digital root sequence (also period-24). (Golden Ratio - Articles)

The first 1000 prime numbers are silently screaming: “Pay attention to us, for we hold the secret to the distribution of all primes!” We heard the call, and with ‘strange coincidences’ leading the way have discovered compelling evidence that the 1000th prime number, 7919, is the perfectly positioned cornerstone of a mathematical object with highly organized substructures and stunning reflectional symmetries. (PrimesDemystified)

I wondered if that property might hold for the incremental powers of phi as well. For this reason I chose to see numbers in the hexagon as quantum, and truncate off the decimal values to determine which integer cell they land in. That is what I found. Phi and its members have a pisano period if the resulting fractional numbers are truncated. (Prime Hexagon).

One of the most promising attempts to go beyond the standard model of particle physics is superstring theory.

• As it is well known, special relativity fused time and space together, then came general relativity and introduced a curvature to space-time.
• Kaluza and later on Klein added one more dimension to the classical four in order to unify general relativity and electromagnetism.

The dimensionality of space-time plays a paramount role in the theoretical physics of unification and has led to the introduction of the 26 dimensions of string theory, the 10 dimensions of superstring theory, and finally the heterotic string theory with the dimensional hierarchy 4, 6, 10, 16 and 26