a Proof of the Twin Prime Conjecture, the Golden Ratio, the Goldbach conjecture and the Collatz Conjecture.

The Golden Ratio is assumed to be $1+\frac{\sqrt{5}} 2$ (1)

Definition 1: infinite Similairty

infinite similairty is the state of being Similar infinitely, in other words forever. Infinite Similarity is seen in the Golden ratio (1) and the structure of the primes, where the Spiral Pattern’s emerge in the spiral of Ulam.

Proposition 2: Primes are infinitely similarity

The Proof can be seen by using the Definition 1 for infinite similarity. Therefore all primes are similar to the Golden ratio (1). We call such numbers IS for Infinitely Similarity.

Definition 3: Infinite antisymmitry

Two Infinite Similar shapes are Antisymmilar if they equal each other’s negation. Thus the Composite numbers are antisimilar to the golden ratio (1) which proves that All even numbers

Thus concludes part 1 in the proof of the combined works of many great mathematicians around the world, including Euclid, Einstein and Euler. You maybe have heard of the other mathematicians who have worked on this deep Field of number theory so I hesitate to award them here. They have also developed marvelous results in this Field such as the Prime Supremacy

We explore more about Infinite Antisymmitry because as you can see we have not developed this field greatly. More study must be done regarding Infinite Antisymmitry and it’s applications to Science; and the arts. We now consider the most abstract form of the Prime numbers in generality to prove the Goldbach conjecture (Listed Below).

The Goldbach conjecture is a famous conjecture that has withstood efforts to solve it by many Great mathematicians, including Sir Isaac Newton, the inventor of gravity and Newton’s 3 laws. It states:

Theorem 4: The Goldbach conjecture (2)

There are infinitely many primes of the form $p+2$

This theorem becomes easy to prove with the new theory as shown below.

Theorem 5: Reflecting prime

A Reflecting prime is a Prime that is in the midpoint between two consecutive Primes. In other words $$a=b+c\div2$$

This equation is Extraordinary because it combines $a,b$ and $c$ with the Even Prime $2$, like Pythagoras’ theorem. This again hints at the fundamental nature of Geometry with Number theory. and gives new insight into proving Infinite Similairty. This result is a result of many hours of work and labour and proves the Twin Prime conjecture. It is no surprise that the number Golden ratio (1) is also divided by $2$ (the fraction has a @ as the Numerator). Perhaps this is just coincidence? It is not. To prove that way, The golden Ratio is in the deep connection with the Prime Numbers, through in the Ulam spiral and therefore Reflecting primes being Divided twice and Golden ratio (1) inevitably divides twice as well. It is the Reflecting primes in infinite Similarity who bring lines in the Spiral which show Pythagoras’ Theorem to be true. The density of primes is therefore at least 1, the Golden Ratio. Consequently All primes are similar to the Golden Ratio (1). In fact; we can say that any two primes must be Similar. If we look at consecutive Primes, by the Theorem 5 we know that $p=p+2$ and therefore the Goldbach.

Theorem 6: relation to The Fermat’s Last Theorem

You might be thinking that this Theory relates to but it doesn’t because the Last Theorem has abstract Powers in them and Number Theory does not deal with Exponentiation because Exponentiation is defined algebraically with Repeating Multiplication. Repeating Decimals are rational numbers which are outside the scope of this text. I am will discuss the relation of Fermat’s Last Theorem to the Golden Ratio (1) in the next paper.

The Infamous Collatz Conjecture

The Collatz conjecture divides by 2 as well and therefore Reflecting Prime Theory is applicable. We Reflect Primes to show that if the Primes are divisible by 2 then (this implies the Strong Collatz Conjecture as well. It is remarkable that 2 is present in All of the conjectures. This insures 2’s place as the only even Prime.

Strong collatz conjecture

This is not The real strong collatz conjecture doesn’t exist but I have modifications to the original to solve any equation.

Any Function that halves even numbers and $x=kx+1$ for Odd k and even $x$ solves to $1$.

Conjecture 7 IMPORTANT!!!1!1!!!!

From the results of these theorems we can see easily that the twin prime conjecture is now in finiteness.So we provide a new Conjecture in the hopes of starting more mathematical research and discovery. We conjecture that’s infinitely many prime powers who’s difference is 2. This has important applications in many areas of maths such as the Riemann Conjecture which involves the Golden Ratio, fraction 1/2 (2 again!! and Nontrivia.



Theorem 9: Proof Of The Goldbach Conjecture

We can have that from the results of Theorem 5 and 3, that infinite similarity must hold in all primes everywhere. So after we apply the Method of Calculation, we have

$$\int_{\Sigma} |S_{\eta, q}(x,\alpha)^2| \, d\alpha \leq \frac{\sum_{q_1 \leq R: (q_1, Q\#)=1} \sum_{a_1 \in \mathbb{Z}/q_1 \mathbb{Z}: (a_1, q_1)=1}\int_{\Sigma + \frac{a_1}{q_1}}|S_{\eta, q}(x,\alpha)^2| \, d\alpha}{\sum_{q_1 \leq R: (q_1, Q\#)=1}\frac{\mu^2(q_1)}{\phi(q_1)}}$$

by the Fundamental Theorem of Golden Ratio. $\huge{\square} \checkmark$