**Theorem of prime numbers**

A 2,500-year-old ratio problem searches for the “hidden”, invisible, infinite ratios of prime numbers, which describe the real structure, and measure their asymptotic distribution over the set of natural numbers.

**“Hidden” and “invisible” ratios in the sense of not immediately apparent.**

The new mathematical structure, the new dynamic system of numbers (ratios ) that is created, and the new mathematical terms that describe the real structure and measure their asymptotic distribution over the set of natural numbers.

**FUNDAMENTAL NUMBER THEOREM**

The fundamental theorem of arithmetic defines the key role of prime numbers in creating the coherent structure of infinite natural numbers.

The Fundamental Theorem of Arithmetic can be defined by two different approaches or formulations, which describe the uniqueness of the representation of every positive integer as a product of prime numbers.

Both approaches express the idea of singularity of representation but with different emphases. Combined with the definition of prime numbers, they create the corresponding mathematical structure of the natural numbers.

The wording of the TTHA is not neutral. It shapes the mathematical structure of the natural numbers in exile.

Therefore we have two different mathematical structures in the natural numbers, which should, with the corresponding mathematical terms of the first (multiples, common multiples…) or the new ones that we have to create for the second, interpret and measure the corresponding properties of the two formulations.

Since these are two different mathematical structures in natural numbers, the mathematical terms that interpret and measure different properties will be different and not the same for both formulations.

One formulation “abolishes” the mathematical terms of the other.

The first formulation of the fundamental theorem of arithmetic which prevailed in the middle of the 20th century, creates in exile the well-known specific coherent structure of natural numbers.

**First wording:**

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

The definition of the prime creates the obvious unique exponent ratios of the primes 1/2, 1/3, 1/5, and 1/7 ..which measure exactly the number of terms in the numerical progression of their multiples.

The mathematical structure of the natural numbers, which is defined in this formulation by the Fundamental Theorem of Arithmetic (FFA), is described and measured by mathematical terms such as multiples, common multiples, least common multiples (LCMs), and the** rational coefficients of the multiples of prime numbers**

**“Every positive integer greater than 1 can be written as a product of one or more prime factors uniquely, except in the order of the factors”.**

**The mathematical structure of the natural numbers, defined by this formulation, is described and measured in mathematical terms as they were first formulated by Euclid.**

__The mathematical terms that describe the mathematical structure__

*Natural numbers.**Prime numbers.**Composite numbers.**Multiples of prime numbers.**Common multiples of prime numbers.**Least common multiples.*

__The mathematical terms that measure the mathematical structure__

Mathematical terms that measure mathematical structure. The rational

coefficients of the multiples of prime numbers, measure exactly the number

of terms in the numerical progression of their multiples.

This banishment formulation does not generate the structure that describes

and measures the asymptotic distribution of primes.

**It formulates and describes the problem and not its solution.**

**From cohesion and collectivity to sharing and uniqueness.**

This formulation of the fundamental theorem of arithmetic has occupied mathematical science throughout time from antiquity to the present 2500 years.

From Euclid and Lagrange, Gauss, to modern mathematicians.

We arrange the prime numbers in ascending order… This form of factorization can be called an abbreviated product form. Wikipedia:

*“Every positive integer greater than 1 can be written as a product of one or more prime factors uniquely, only according to the ascending order of the factors”.*

This formulation of the Fundamental Theorem of Arithmetic creates by default a new specific proportional mathematical structure in the natural numbers.

The same mathematical structures (same ratios) are not created and the mathematical terms of the other formulation (multiples, common, least common multiples) and the ratios of prime numbers, the explicit coefficients for measuring the numerical progress of their multiples, cannot be interpreted and measure the new structure that is created.

The basic mathematical structure of the other formulation is removed.

The fundamental questions were either not formulated or the answers were not found.

•* What specific new mathematical structure is created by this formulation?*

**• What mathematical term describes and measures these properties in the new dynamic system of numbers (proportions) created?**

• *What are the new rational ratios of the prime numbers that count this new class of their multiples that count only once with the uniqueness of their representation and their factorization?*

•** The prime number theorem to be proved approximately relied on the mathematical structure of only the first formulation.**

What basic mathematical structure of the prime number theorem is necessarily created by this formulation of the ΘΤΑ?

What can the newly created dynamical number system bring to understanding the structure, precise description, and measurement of the asymptotic distribution of prime numbers?

The new mathematical terms describe the new ratios of primes that are created and measure the uniqueness of the factorization of complex numbers.

*“Every positive integer greater than 1 can be written as a product of one or more prime factors uniquely, only according to the ascending order of the factors.”*

The mathematical structure created on the set of banishing natural numbers by this formulation has only three mathematical terms to describe and measure these particular properties.

**The mathematical terms that describe the mathematical structure:**

The mathematical structure that the mathematical terms must describe has not yet been found and formulated.

*Natural numbers.**Prime numbers.***Composite numbers (multiples of prime numbers) are counted only once by the uniqueness of their representation as a product of prime numbers according to the ascending order of arrangement of the factors**.*They are unique multiples of prime numbers.*

**The mathematical terms that measure the mathematical structure:**

**The new mathematical structure created on the set of banishing natural numbers by this formulation has not yet been described.**

**The corresponding mathematical terms that describe and measure these properties have not been formulated.**

In the next post:

**PRIME NUMBER THEOREM: “The hidden ratios”.**

*The fundamental rule that with its formulation creates a new mathematical structure, the new dynamic system of numbers (ratios).*

*The new mathematical terms describe the real structure and measure with the new ratios of the prime numbers their asymptotic distribution in the set of natural numbers.*