Medical, Pharma, Engineering, Science, Technology and Business

Institute for Basic Research, Palm Harbor, P. O. Box 3924, Beijing 100854, P. R. China

- *Corresponding Author:
- Jiang CX

Institute for Basic Research

Palm Harbor, P. O. Box 3924

Beijing 100854, P. R. China

**Tel:**+1-727-688 3992

**E-mail:**[email protected]

**Received Date**: March 10, 2017; **Accepted Date:** April 17, 2017; **Published Date**: April 27, 2017

**Citation: **Jiang CX (2017) On the Fundamental Theorem in Arithmetic Progression
of Primes. J Generalized Lie Theory Appl 11: 264. doi: 10.4172/1736-4337.1000264

**Copyright:** © 2017 Jiang CX. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

Using Jiang function we prove the fundamental theorem in arithmetic progression of primes. The primes contain only *k<P _{g}*+1 long arithmetic progression, but the primes have no

Arithmetic; Lie theory; Fundamental theorem; Progression; Asymptotic formula

The fundamental theorem in arithmetic progression of primes.

We define the arithmetic progression of primes [1-3].

(1)

Where is called a common difference, Pg is called g-th prime.

(2)

X(P) denotes the number of solutions for the following congruence:

(3)

Where q=1,2,…,P−1.

If then X(P)=0;X(P)=k−1 otherwise. From eqn (3) we have:

(4)

If k=P_{g+1} then J_{2}(P_{g+1})=0, J_{2}(ω)=0, there exist finite primes P_{1} such that P_{2},…,P_{k} are primes. If k<P_{g+1} then J_{2}()≠0, there exist infinitely many primes P1 such that P_{2},…,P_{k} are primes. The primes contain only k<P_{g+1} long arithmetic progression, but the primes have no k>P_{g+1} long arithmetic progression geometry. We have the best asymptotic formula [1-3]:

(5)

Where is called primorial, φ(ω) Euler function.

Suppose k=P_{g+1}−1. From eqn (1) we have:

P_{i+1}=P_{1}+ω_{g}i,i=0,1,2,…,P_{g+1}2. (6)

(7)

We prove that there exist infinitely many primes P1 such that are primes for all P_{g+1}.

From eqn (5) we have:

(8)

From eqn (8) we are able to find the smallest solutions for large P_{g+1}.

Theorem is foundation for arithmetic progression of primes.

**Example 1:** Suppose P_{1}=2, ω_{1}=2, P_{2}=3. From eqn (6) we have the twin primes theorem:

P_{2}=P_{1}+2. (9)

From eqn (7) we have:

(10)

We prove that there exist infinitely many primes P_{1} such that P_{2} are primes. From eqn (8) we have the best asymptotic formula [4-6]:

(11)

Twin prime theorem is the first theorem in arithmetic progression of primes.

**Example 2:** Suppose P_{2}=3, ω_{2}=6, P_{3}=5. From eqn (6) we have:

(12)

From eqn (7) we have:

(13)

We prove that there exist infinite many primes P_{1} such that P_{2}, P_{3} and P_{4} are primes. From eqn (8) we have the best generalized asymptotic formula:

(14)

**Example 3:** Suppose P_{9}=23, ω_{9}=223092870, P_{10}=29. From eqn (6) we have:

(15)

From eqn (7) we have:

(16)

We prove that there exist infinitely many primes P1 such that P_{2},…, P_{28} are primes. From eqn (8) we have the best asymptotic formula [7]:

(17)

From eqn (17) we are able to find the smallest solutions π_{28}(N_{0},2)>1.

On May 17, 2008, Wroblewski and Raanan Chermoni found the first known case of 25 primes:

61711054912832631+366384×ω_{23}×n, for n=0 to 24.

Theorem can help in finding for 26, 27, 28,…, primes in arithmetic progressions of primes.

**Corollary 1:** Arithmetic progression with two prime variables.

Suppose ω_{g}=d. From eqn (1) we have:

P_{1},P_{2}=P_{1}+d,P_{3}=P_{1}+2d,…,P_{k}=P_{1}+(k−1)d, (P_{1},d)=1 (18)

From eqn (18) we obtain the arithmetic progression with two prime variables: P_{1} and P_{2},

P_{3}=2P_{2}−P_{1}, P_{j}=(j−1)P_{2}−(j−2)P_{1}, 3 ≤ j ≤ k < P_{g+1}. (19)

We have Jiang function [3]:

(20)

X(P) denotes the number of solutions for the following congruence matrices:

(21)

where q_{1}=1,2,…,P−1; q2=1,2,…,P−1.

From eqn (21) we have:

(22)

We prove that there exist infinitely many primes P_{1} and P_{2} such that P_{3},…, P_{k} are primes for 3≤k<P_{g+1}.

We have the best asymptotic formula [8]:

(23)

From eqn (23) we have the best asymptotic formula:

(24)

From eqn (24) we are able to find the smallest solution π_{k−1}(N_{0},3)>1 for large k<P_{g+1}.

**Example 4.** Suppose k=3 and P_{g+1} >3. From eqn (19) we have:

P_{3}=2P_{2}−P_{1}. (25)

From eqn (22) we have:

(26)

We prove that there exist infinitely many primes P1 and P2 such that P_{3} are primes. From eqn (24) we have the best asymptotic formula:

(27)

Example 5: Suppose k=4 and P_{g+1} >4. From eqn (19) we have:

P_{3}=2P_{2}−P_{1}, P_{4}=3P_{2}−2P_{1}. (28)

From eqn (22) we have:

(29)

We prove that there exist infinitely many primes P_{1} and P_{2} such that P_{3} and P_{4} are primes. From eqn (24) we have the best asymptotic formula:

(31)

**Example 6:** Suppose k=5 and P_{g+1} >5. From eqn (19) we have:

P_{3}=2P_{2}−P_{1}, P_{4}=3P_{2}−2P_{1}, P_{5}=4P_{2}−3P_{1}. (31)

From eqn (22) we have:

(32)

We prove that there exist infinitely many primes P_{1} and P_{2} such that P_{3}, P_{4} and P_{5} are primes. From eqn (24) we have the best asymptotic formula:

(33)

**Corollary 2:** Arithmetic progression with three prime variables.

From eqn (18) we obtain the arithmetic progression with three prime Lie Theory variables: P_{1}, P_{2} and P_{3}.

(34)

We have Jiang function:

(35)

X(P)denotes the number of solutions for the following congruence:

(36)

Where q_{i}=1,2,…,P−1, i=1,2,3.

**Example 7:** Suppose k=4 and P_{g+1}>4. From eqn (34) we have:

P_{4}=P_{3}+P_{2}−P_{1}. (37)

From eqns (35) and (36) we have:

(38)

We prove that there exist infinitely many primes P_{1} and P_{2} and P_{3} such that P_{4} are primes. We have the best asymptotic formula:

(39)

For k≥5 from eqns (35) and (36) we have Jiang function:

(40)

We prove that there exist infinitely many primes P_{1} and P_{2} and P_{3} such that P_{4},…, P_{k} are primes for 5 ≤ k≤P_{g+1}.

We have the best asymptotic formula:

(41)

From eqn (41) we have:

(42)

From eqn (42) we are able to find the smallest solution π_{k−2}(N_{0},4)>1 for large k<P_{g+1}.

**Corollary 3:** Arithmetic progression with four prime variables.

From eqn (18) we obtain the arithmetic progression of algebra with four prime variables: P_{1}, P_{2}, P_{3} and P_{4}

P_{5}=P_{4}+2P_{3}−3P_{2}+P_{1}, Pj=P_{4}+(j−3)P_{3}−(j−2)P_{2}+P_{1}, 5 ≤ j ≤ k < P_{g+1} (43)

We have Jiang function:

(44)

X(P) denotes the number of solutions for the following congruence:

(45)

Where,

q_{i}=1,…,P−1,i=1,2,3,4

**Example 8:** Suppose k=5 and k<Pg+1>5. From eqn (43) we have:

P_{5}=P_{4}+2P_{3}−3P_{2}+P_{1}. (46)

From eqns (44) and (45) we have:

(47)

We prove there exist infinitely many primes P_{1}, P_{2}, P_{3} and P_{4} such that P_{5} are primes.

We have the best asymptotic formula:

(48)

**Example 9:** Suppose k=6 and P_{g+1} >6. From eqn (43) we have:

P_{5}=P_{4}+2P_{3}−3P_{2}+P_{1}, P_{6}=P_{4}+3P_{3}−4P_{2}+P_{1}. (49)

From eqns (44) and (45) we have:

(50)

We prove there exist infinitely many primes P_{1}, P_{2}, P_{3} and P_{4} such that P_{5} and P_{6} are primes.

We have the best asymptotic formula:

(50)

For k≥7 from eqns (44) and (45) we have Jiang function:

(51)

We prove there exist infinitely many primes P_{1}, P_{2}, P_{3} and P_{4} such that P_{5},…,P_{k} are primes.

We have best asymptotic formula:

(52)

I thank Professor Huang Yu-Zhen for computation of Jiang functions.

- Jiang CX (1995) On the prime number theorem in additive prime number theory, Preprint, 1995.
- Jiang CX (2006) The simplest proofs of both arbitrarily long arithmetic progressions of primes.Preprint.
- Jiang CX (2002) Foundations of Santiili’sisonumber theory with applications to new cryptograms, Fermat’s theorem and Goldbach’s conjecture, Inter. Acad. Press, 68-74, 2002, MR 2004c: 11001.
- Green B, Tao T (2008)The primes contain arbitrarily long arithmetic progressions.Ann Math 167: 481-547.
- Tao T (2007)The dichotomy between structure and randomness, arithmetic progressions, and the primes. In: Proceedings of the international congress of mathematicians (Madrid), Europ Math Soc 1: 581-609.
- Tao T, Vu V (2006) Additive combinatorics. Cambridge University Press.
- Tao T (2006) Long arithmetic progressions in the primes. Australian mathematical society meeting, 10.
- Tao T (2007)What is good mathematics? Bull AmerSoc 44: 623-634.

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