A modeling of integers with prime pairs and its reversed extension and models

Yucong Duan

DIKW research group, Hainan University, China

Email: duanyucong@hotmail.com

DIKW (yucongduan.org)

Abstraction:

We propose an expression mechanism from classic series of integers to series of primes in the forms of intervals as basis. We comprise integers with seamless prime pairs, and comprise integers with seamless prime pairs. Thereafter we express even decomposition integers with primes. We further propose potentiality useful conclusions on interesting models. We end with a philosophical and esthetic retrospection on this topic.

1. From classic series of integers to series of primes in the forms of intervals as basis

Denote prime series P as {p₁, p₂, p₃, ..., p_n-1, p_n}

Denote the integer difference between p_m and p_m+1 as

dp_m=p_m+1 -p_m

The DP series is denoted as {dp₁, dp₂, dp₃, ..., dp_n-1, dp_n}

If we start from a prime p_x then p_x+1 can be expressed as:

p_x+1=p_x+dp_x

Extensional conclusion:

Any prime p_x+m can be expressed seamlessly as:

p_x+m=p_x+dp_x+1+dp_x+2+dp_x+3+...+dp_m-1+dp_m

2. Comprising integers with seamless prime pairs

We can assume that we have verified a seamless series of even integers of EV_h= {ev₁, ev₂, ev₃, ..., ev_h-1, ev_h} with every ev can be expressed at least as addition of a pair of primes from P_h.

ev_{EVh}=((p_a + p_b )_{Ph})_|{(a,b)}|>=1

Suppose that all even number in EV_h has been validated as conforming to

ev_{EVh}=((p_a + p_b )_{Ph})_|{(a,b)}|>=1.

If an even number ev_h+1 cannot be expressed as addition of 2 p, it can be expressed as:

 ev_h+1!=p_y + p_z , p_y , p_z belongs to P_h. Or it can be alternatively expressed as:

ev_h+1!=(p_y + p_z )_{Ph} 

Or it can expressed as:

ev_{EV(h+1)}=((p_a + p_b )_{Ph})_|{(a,b)}|=0

For every even number ev in EV_h= {ev₁, ev₂, ev₃, ..., ev_h-1, ev_h}, there exists an seamless series of integer number int in INT_h= {int₁, int₂, int₃, ..., int_(h-1)/2, int_h} by subtracting the ev with 2.

According to ev_{EVh}=((p_a + p_b )_{Ph})_|{(a,b)}|>=1, every int in INT_h can be transitively mapped to at least 1 pair of prime in the form of

 {(p_a , p_b ) }_{Ph} or {(p_a , p_b )₁, (p_a , p_b )₂,(p_a , p_b )₃,..,(p_a , p_b )_h-1,(p_a , p_b )_h }_{Ph} .  

Since  INT_h= {int₁, int₂, int₃, ..., int_(h-1)/2, int_h} represent seamless series of integers from int₁ to int_h,  {(p_a , p_b ) }_{Ph} or {(p_a , p_b )₁, (p_a , p_b )₂,(p_a , p_b )₃,..,(p_a , p_b )_h-1,(p_a , p_b )_h }_{Ph} map to seamless series of integers from int₁ to int_h as well.

3. Expressing even decomposition integers with primes

If ev_h+1!=(p_y + p_z )_{Ph} is founded, it can be concluded that:

ev_h+1

!=(p_y + p_z )_{Ph}

=(p_y + !p_z )_{Ph}

It means that starting from any (p_y)_{Ph} , by increasing seamlessly or decrease seamlessly, (!p_z )_{Ph} is continuous within the limit of P_h. This conclusion contradict with the fact that the series of (!p_z )_{Ph} and (p_y)_{Ph}  series intersect at least once in P_h.  

4. Potentiality useful conclusions on interesting models

1. Every integers can be mapped to at least a pair of primes (p_a , p_b ).

2. Each integer can be viewed as constructed as the positive length scope of (-p_a , p_b ) geometrically.

3. Among all the prime pairs mapping to an even number ev, taking the number of ev as the center, using the subtraction and addition by 2, a prime pair can be reached. This series of prime is the continues series of prime pairs which can be used to identify each even with uniqueness as a whole.

4. {(p_y - p_z )_{Ph}}_unique of unique expression for each even, can construct a seamless integer series.  

5. Integers can be viewed not as composing primes but reversely composed by primes.

6. The minimum intervals of integer numbers can be viewed as not only as “1” but also as piece of intervals between two primes, |(p_y - p_z )|_{Ph}.

7. Primes instead of integers can model numbers more essentially or in the extreme essence.

5.  Philosophical and esthetic retrospection

The demonstration of the decomposing need be based on existing series and the seamlessness of the existing decomposition. The symmetric decomposition from the center in the form of prime intervals which is seamless can help get the intuition of the understanding. The decomposing analysis doesn’t really involves classic integer arithmetic calculation but instead relying on the consistency and seamless of series and the mapping and transitivity among the series .  

Reference:

There is no open reference to outside material.

 (PDF) A modeling of integers with prime pairs and its reversed extension and models (researchgate.net)