** The big pattern of prime numbers**

**1. The definition of the prime creates the obvious ratio relations of the infinite primes and the set of infinite natural numbers.**

**2. The formulation of the definition of the fundamental theorem of arithmetic creates the coherent structure “the pattern” of infinite apparent ratios of numbers, the coherence and properties of an infinite set of number relations, such as the natural numbers. armonikos.gr©**

“**The solution to the prime number theorem lies in the “mysterious,” hidden ratios of prime numbers, which have not yet been revealed”**

**ΘΕΩΡΗΜΑ ΤΩΝ ΠΡΩΤΩΝ ΑΡΙΘΜΩΝ : ΕΝΑ ΠΡΟΒΛΗΜΑ ΕΞ ΟΡΙΣΜΟΥ**.

**PART 1-5 ^{ο}**

**PRIME NUMBER THEOREM A PROBLEM BY DEFINITION**

The main reason for formulating a definition, a number theorem, is to create the structure, the properties, and the ratios that create the relevant relations for the coherent structure of the set of numbers, and after creating the structure of the set, it can interpret it according to the properties specified by the wording of the definition.

The theorems, rules, and definitions that make up and try to explain the theorem of prime numbers, as they are formulated and interpreted, offer the possibility of understanding and solving or may make it difficult to solve and understand the distribution of prime numbers. Thus reinforcing the “conjecture”, that the prime number theorem may also be a problem of formulation and interpretation of definitions.

Humans can recognize and understand numbers and their proportions. This allows us to measure, plan, and create.

The total theorem of prime numbers is chosen to be formulated and interpreted in this logical order of basic definitions.

1. The definition of the prime creates the obvious ratio relations of the infinite primes and the set of infinite natural numbers.

2. The formulation of the definition of the fundamental theorem of arithmetic establishes the coherent structure of the infinite apparent ratios of numbers, the coherence and properties of an infinite set of number relations, such as the natural numbers.

3. The asymptotic rule is the one that describes the structure and asymptotic distribution of prime numbers.

4. The way this structure is described by the asymptotic rule is with the logarithmic distribution of primes.

5. The fundamental theorem of arithmetic is an important definition, but it cannot give us all the information we need to understand prime numbers.

6. With this formulation it does not provide a way to count or calculate the percentage of complex numbers, it does not provide a way to count the number of prime numbers.

7. The formulation of a definition in mathematics cannot give us more information about the relationships of numbers than the basis of the definition creates.

The specific formulation and interpretation of the definitions for the distribution of prime and complex numbers led us to an excellent approximate mathematical proof for the distribution of prime and complex numbers. However, since it is “approximate” it may have led us to a misleading understanding of the true structure of the primes and wrong assumptions about the behavior of the primes.

The formulation and interpretation of mathematical definitions affect our understanding and approach to a mathematical subject.

**Can the prime number theorem be a problem by definition?**

**PART 2-5**^{ο}^{}

**FUNDAMENTAL THEOREM OF ARITHMETIC**

Humans can recognize and understand numbers and their proportions. This allows us to measure, plan, and create.

The Fundamental Theorem of Arithmetic and the definition of the prime number establish by definition the obvious ratios of primes and natural numbers.

The formulation chosen in the fundamental theorem of arithmetic:

Every positive integer of 1 can be written as a product of one or more primes uniquely except in the ordering order of the factors that are prime numbers. (without regard to the order of the factors).

The specific wording and interpretation of the definitions for the distribution of prime numbers is not wrong. Carefully selected.

1. The specific formulation and interpretation of the definitions for the distribution of prime and complex numbers constitute a comprehensive mathematical approach. It has been carefully selected to present clearly and precisely the mathematical concepts and theorems related to the coherent structure of primes and natural numbers.

2. The fundamental theorem of arithmetic determines with its specific formulation the coherent structure of the set of infinite natural numbers.

The enormous importance of prime numbers for number theory and mathematics in general stems from the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic defines the key role of prime numbers in number theory:

In mathematics, a prime number is a natural number with the property that its only natural divisors are unity and itself.

Prime numbers can thus be considered basic building blocks of natural numbers.

This formulation of the definitions was chosen because

The definition of the prime creates the apparent unique ratios of the primes 1/2, 1/3, 1/5, and 1/7, and the fundamental theorem of arithmetic determines by its formulation the coherent structure of the set of infinite natural numbers that are described in the form of multiples and common multiples of every prime number.

1. They create the explicit ratios of prime numbers .1/2, 1/3, 1/5……

2. They create the numerical progressions of multiples of infinite natural numbers in the same way. Each natural number multiplied by the infinite natural numbers based on its ratio creates the numerical progression of its multiples.

3. The multiples of every natural number is a function (numerical progression) with the asymptotic distribution of its terms.

4. The rational factor of the ratio of any natural number measures exactly the numerical progress of its multiples.

5. It creates the class of common multiples of prime numbers that connects complex numbers to prime numbers, thus providing the coherent structure of ratio in the set of natural numbers.

**Can the prime number theorem be a problem by definition?**

**PART 3-5**^{ο}^{}

**FUNDAMENTAL THEOREM OF ARITHMETIC**

The wording:

Every positive integer of 1 can be written as a product of one or more primes uniquely except in the ordering order of the factors that are prime numbers. (without regard to the order of the factors).

With this formulation, it also creates the problem of definition:

Complex numbers are created by common multiples of prime numbers. The uniqueness of the primes and the multiplicity of the complex numbers are reconciled with the coherent structure of their common ratio and create the totality of the ratio of the infinite natural numbers.

1. The fundamental theorem of arithmetic is an important definition, but it cannot give us all the information we need to understand prime numbers.

2. With this formulation it does not provide a way to count or calculate the percentage of complex numbers, it does not provide a way to count the number of primes

3. The main conclusion is that the formulation of a definition cannot give us more information about the relations of numbers than it can create.

It cannot count the common multiples of prime numbers only once thus constituting the error of counting the actual number of complex numbers in the set of natural numbers.

Common multiples of prime numbers are also simple multiples of every prime number.

This formulation of the definitions was chosen because it reconciles the uniqueness of the ratios of the primes with the common multiplicity of the ratios of the complex numbers and thus creates the coherent structure of the totality of the ratio of the infinite natural numbers.

For simple and practical mathematical reasons, e.g.

They make the four basic arithmetic operations simple to understand and easy to use.

All complex numbers and multiples of natural numbers are created in the same simple way, infinite natural numbers times infinite natural numbers.

This formulation in the fundamental theorem of arithmetic was chosen to define in the same simple way the creation of an infinite but coherent structure of natural numbers based on the unique ratios, and multiples of the prime numbers. Creating the common multiples which are also atomic multiples of the corresponding prime numbers. Reconciling the competing prime factors of every complex number that is a common multiple of them.

We could also call it the fundamental theorem of the natural numbers as its formulation prioritizes the creation of the coherent structure of the natural numbers, reconciling the competing unique ratios of the prime numbers into a common proportional infinite set of complex numbers and not the measurement of Prime numbers.

**Can the prime number theorem be a problem by definition? **

** Α.**** ****Table of arithmetic progressions of multiples of prime**** ****numbers 1-31 from 1-100.**

** ****B. Table: Composite numbers (pink color). **

** ****B. Table: Prime numbers (green color).**

** ****Error****.**** ****Measurement error of composite numbers equal to ****20****.**

** ****common multiples****: ****20**** **** **

**PART 4-5 ^{ο}**

**ASYMPTOTIC DISTRIBUTION RULE**

In number theory, the prime number theorem describes the asymptotic distribution of prime numbers among the positive integers.

1. The fundamental theorem of arithmetic is an important definition, but it cannot give us all the information we need to understand prime numbers.

2. With this formulation it does not provide a way to count or calculate the percentage of complex numbers, it does not provide a way to count the number of primes

3. With this formulation it is by definition a problem of counting prime numbers.

Choice of interpretation:

The asymptotic distribution rule examines the behavior of a potential variable for large values of the sample size. It is a useful tool, but it does not give us a complete picture of the true structure of the asymptotic distribution of primes.

However, there are cases where the asymptotic approximation can be misleading regarding a function’s true structure or behavior at specific points or regions.

For example, the asymptotic distribution rule of prime numbers is that their density approaches the Dirichlet frequency distribution with parameters (1/2, 1/2, 1/2, …, 1/2). This means that prime numbers are evenly distributed on a logarithmic scale.

The actual structure of prime numbers may be much more complicated than the simple uniform distribution described by the asymptotic distribution rule.

The asymptotic rule is general to the asymptotic distribution of primes without being able to describe it, hence the “randomness” and periodic patterns that coexist in the structure of prime numbers cannot be explained.

The specific formulation and interpretation of the definitions in the fundamental theorem of arithmetic for the distribution of prime and complex numbers led us to an excellent approximate mathematical proof for the distribution of prime and complex numbers. However, since it is “approximate” it may have led us to a misleading understanding of the true structure of the primes and wrong assumptions about the behavior of the primes.

Could this particular interpretation of the definition of the asymptotic distribution rule of primes have led us to a misleading approximation of the real behavior of the structure of primes?

We usually refer to the asymptotic distribution of only the prime numbers, which are the tiny percentage to the “infinitely maximal” percentage of the complex numbers that make up the infinite set of natural numbers.

Does the structure of the distribution of prime numbers need to be integrated into the whole structure of the natural numbers to be understood? The primes generate the asymptotic distribution of the complex numbers with their common multiples and essentially the asymptotic distribution on the largest primes.

**Can the prime number theorem be a problem by definition? **

**PART 5-5**^{Ο}** **

**Logarithmic Distribution of Prime Numbers**

The asymptotic law is the one that describes the structure and distribution of prime numbers. And the way that this structure is described by the asymptotic law is with the logarithmic distribution of primes.

The official view of the mathematical community for why the decrease in the percentage of prime numbers gradually slows down is that this is due to the logarithmic distribution of prime numbers.

The logarithmic distribution of prime numbers means that the number of prime numbers that are less than or equal to a certain number x increases logarithmically with x. This means that the number of prime numbers increases more quickly in the first intervals of natural numbers, but slows down gradually in larger intervals.

The asymptotic distribution law is an approximation, a useful tool for the distribution of prime numbers for large values of the sample size, but it does not give us a complete picture of the real structure of the asymptotic distribution of primes.

In some cases, the asymptotic approximation can be misleading about a function’s real structure or behavior at specific points or regions. This can happen because the approximation is based on certain assumptions that may not hold for specific cases.

The logarithmic distribution of primes is how this structure is described by the asymptotic law.

It is important to note that the logarithmic distribution is only an approximation of the real distribution of prime numbers. It is not an accurate description of the distribution of prime numbers for all values of the number.

In the case of the distribution of prime numbers, the logarithmic distribution is based on the assumption that prime numbers are uniformly distributed in a given range. However, the real distribution of prime numbers appears to be more concentrated in certain regions. This can lead to errors in the asymptotic approximation.

Then the logarithmic distribution can be misleading about the real structure or behavior of the distribution of primes.

Theorems and results within analytic number theory tend not to be exact structural results about integers, for which algebraic and geometric tools are more appropriate. Instead, they give approximate bounds and estimates for various theoretical number functions.

«The prime number theorem has no solution, or it lies in the choice of definitions that formulate the overall prime number theorem differently and create a new structure of ratios that can prove the prime number theorem».

“**The solution to the prime number theorem lies in the “mysterious,” hidden ratios of prime numbers, which have not yet been revealed**“

In the next section of publications and details on my website armonikos.gr . Τhe solution.

**Prime Number Theorem: The “Hidden Ratios”**

**Nikos Giannoulas : armonikos.gr©** ** **