All submissions of the EM system will be redirected to Online Manuscript Submission System. Authors are requested to submit articles directly to Online Manuscript Submission System of respective journal. Liu Yajun*

South China University of Technology, Guangzhou P.R, 510640, China, Email: yajun@scut.edu.cn

*Correspondence: Liu Yajun, South China University of Technology, Guangzhou P.R, 510640, China, Email: yajun@scut.edu.cn

Received: 27-Dec-2022, Manuscript No. puljpam-22-6002; Editor assigned: 29-Dec-2022, Pre QC No. puljpam-22-6002 (PQ); Accepted Date: Jan 03, 2023; Reviewed: 31-Dec-2022 QC No. puljpam-22-6002 (Q); Revised: 02-Jan-2023, Manuscript No. puljpam-22-6002 (R); Published: 30-Jan-2023, DOI: 10.37532/2752-8081.23.7(1).20-21.

Citation: Yajun L. The symmetry of N-domain and prime conjectures. J Pure Appl Math. 2023; 7(1):20-21.

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## Abstract

In this paper, we discuss the symmetry of N-domain and we find that using the symmetry characters of Natural Numbers we can give proofs of the Prime Conjectures: a concise proof of Fermat last theorem Polignac’s conjecture (twins Prime Conjecture) Goldbach Conjecture and Reimann Hypothesis.

### Keywords

N domain; Prime conjectures

### Introduction

We have

N ~（ 0,1,2,3,4,...........） all the natural numbers

n ~（ 1,2,3,4,...........） all the natural numbers excepted 0

P ~（ 2,3,5,7,...........） all the prime numbers

p ~（ 3,5,7,...........） all the odd prime numbers

We notice that

N ~（ 0, n）

P ~（ 2, p)

1: The proof of twin primes conjecture and goldbach conjecture

Figure 2: All the non-trivial Zero points of Riemann zeta-function are on the 1/2 axis

We can define a N, n, P, p, 2n coordinate system shown in Figures 1 and 2.  p1, p0, p2 ∈ p

we can get

p1 → n - 1

p0 → n

p2 → n + 1

And

- p1 → - (n-1)

- p0 → -n

- p2 → - (n+1)

So we have

p2 + (- p1) → n + 1 - (-(n-1)) = 2n

This is the proof of Polignac’s conjecture.

And

p2 - p1 → (n+1) - (n-1) = 2

This is the proof of twin primes conjecture.

And

2n = n + 1 + n - 1 → p2 + p1

And

n -1 > 2 n > 3 so 2n > 6

This mean that every even number bigger than six can be divided into two odd prime numbers in N domain.

This is the proof of Goldbach conjecture.

2: The proof of Riemann Hypothesis

Riemann Zeta-Function is Riemann hypothesis

All the Non-trivial zero-point of Zeta-Function Re(s ) = 1/2 .

We have

0 = 1 / 2 - 1 / 2

1 = 1 / 2 + 1 / 2

i2 = - 1

1 / 2 = 1 / 2 * (1/2+1/2i) (1/2-1/2i) The tr(A)=1/2*n

This is mean that all the non-trivial Zero points of Riemann zetafunction are on the 1/2 axis just show as Fig.3. This is the proof of Hilbert–Pólya conjecture. So we give a proof of Riemann Hypothesis (Figures 2 and 3).

3: A concise proof of fermat’ last theorem

We can definite a function as So x + y = n And x + y ∈ p

x2 + y2 = n2  But we notice that:

When x + y = 3∈p xy = 3 (x = 1, y + 3) This is equal to X3 + Y3 = n3 has no integer solution. This is a concise proof of Fermat’ last Theorem.

## Recommended Conferences

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