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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On the Fundamental Theorem in Arithmetic Progression of Primes

Jiang CX*

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.

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Abstract

Using Jiang function we prove the fundamental theorem in arithmetic progression of primes. The primes contain only k<Pg+1 long arithmetic progression, but the primes have no k>Pg+1 long arithmetic progressions theorem.

Keywords

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

Theorem

The fundamental theorem in arithmetic progression of primes.

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

equation (1)

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

We have Jiang function [1-3]:

equation (2)

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

equation (3)

Where q=1,2,…,P−1.

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

equation (4)

If k=Pg+1 then J2(Pg+1)=0, J2(ω)=0, there exist finite primes P1 such that P2,…,Pk are primes. If k<Pg+1 then J2()≠0, there exist infinitely many primes P1 such that P2,…,Pk are primes. The primes contain only k<Pg+1 long arithmetic progression, but the primes have no k>Pg+1 long arithmetic progression geometry. We have the best asymptotic formula [1-3]:

equation

equation (5)

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

Suppose k=Pg+1−1. From eqn (1) we have:

Pi+1=P1gi,i=0,1,2,…,Pg+12. (6)

equation (7)

We prove that there exist infinitely many primes P1 such that equation are primes for all Pg+1.

From eqn (5) we have:

equation (8)

From eqn (8) we are able to find the smallest solutions equation for large Pg+1.

Theorem is foundation for arithmetic progression of primes.

Example 1: Suppose P1=2, ω1=2, P2=3. From eqn (6) we have the twin primes theorem:

P2=P1+2. (9)

From eqn (7) we have:

equation (10)

We prove that there exist infinitely many primes P1 such that P2 are primes. From eqn (8) we have the best asymptotic formula [4-6]:

equation (11)

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

Example 2: Suppose P2=3, ω2=6, P3=5. From eqn (6) we have:

equation (12)

From eqn (7) we have:

equation (13)

We prove that there exist infinite many primes P1 such that P2, P3 and P4 are primes. From eqn (8) we have the best generalized asymptotic formula:

equation (14)

Example 3: Suppose P9=23, ω9=223092870, P10=29. From eqn (6) we have:

equation (15)

From eqn (7) we have:

equation (16)

We prove that there exist infinitely many primes P1 such that P2,…, P28 are primes. From eqn (8) we have the best asymptotic formula [7]:

equation (17)

From eqn (17) we are able to find the smallest solutions π28(N0,2)>1.

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

61711054912832631+366384×ω23×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 ωg=d. From eqn (1) we have:

P1,P2=P1+d,P3=P1+2d,…,Pk=P1+(k−1)d, (P1,d)=1 (18)

From eqn (18) we obtain the arithmetic progression with two prime variables: P1 and P2,

P3=2P2−P1, Pj=(j−1)P2−(j−2)P1, 3 ≤ j ≤ k < Pg+1. (19)

We have Jiang function [3]:

equation (20)

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

equation (21)

where q1=1,2,…,P−1; q2=1,2,…,P−1.

From eqn (21) we have:

equation (22)

We prove that there exist infinitely many primes P1 and P2 such that P3,…, Pk are primes for 3≤k<Pg+1.

We have the best asymptotic formula [8]:

equation

equation (23)

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

equation (24)

From eqn (24) we are able to find the smallest solution πk−1(N0,3)>1 for large k<Pg+1.

Example 4. Suppose k=3 and Pg+1 >3. From eqn (19) we have:

P3=2P2−P1. (25)

From eqn (22) we have:

equation (26)

We prove that there exist infinitely many primes P1 and P2 such that P3 are primes. From eqn (24) we have the best asymptotic formula:

equation (27)

Example 5: Suppose k=4 and Pg+1 >4. From eqn (19) we have:

P3=2P2−P1, P4=3P2−2P1. (28)

From eqn (22) we have:

equation (29)

We prove that there exist infinitely many primes P1 and P2 such that P3 and P4 are primes. From eqn (24) we have the best asymptotic formula:

equation (31)

Example 6: Suppose k=5 and Pg+1 >5. From eqn (19) we have:

P3=2P2−P1, P4=3P2−2P1, P5=4P2−3P1. (31)

From eqn (22) we have:

equation (32)

We prove that there exist infinitely many primes P1 and P2 such that P3, P4 and P5 are primes. From eqn (24) we have the best asymptotic formula:

equation (33)

Corollary 2: Arithmetic progression with three prime variables.

From eqn (18) we obtain the arithmetic progression with three prime Lie Theory variables: P1, P2 and P3.

equation (34)

We have Jiang function:

equation (35)

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

equation (36)

Where qi=1,2,…,P−1, i=1,2,3.

Example 7: Suppose k=4 and Pg+1>4. From eqn (34) we have:

P4=P3+P2−P1. (37)

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

equation (38)

We prove that there exist infinitely many primes P1 and P2 and P3 such that P4 are primes. We have the best asymptotic formula:

equation (39)

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

equation (40)

We prove that there exist infinitely many primes P1 and P2 and P3 such that P4,…, Pk are primes for 5 ≤ k≤Pg+1.

We have the best asymptotic formula:

equation

equation (41)

From eqn (41) we have:

equation (42)

From eqn (42) we are able to find the smallest solution πk−2(N0,4)>1 for large k<Pg+1.

Corollary 3: Arithmetic progression with four prime variables.

From eqn (18) we obtain the arithmetic progression of algebra with four prime variables: P1, P2, P3 and P4

P5=P4+2P3−3P2+P1, Pj=P4+(j−3)P3−(j−2)P2+P1, 5 ≤ j ≤ k < Pg+1 (43)

We have Jiang function:

equation (44)

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

equation (45)

Where,

qi=1,…,P−1,i=1,2,3,4

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

P5=P4+2P3−3P2+P1. (46)

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

equation (47)

We prove there exist infinitely many primes P1, P2, P3 and P4 such that P5 are primes.

We have the best asymptotic formula:

equation (48)

Example 9: Suppose k=6 and Pg+1 >6. From eqn (43) we have:

P5=P4+2P3−3P2+P1, P6=P4+3P3−4P2+P1. (49)

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

equation (50)

We prove there exist infinitely many primes P1, P2, P3 and P4 such that P5 and P6 are primes.

We have the best asymptotic formula:

equation (50)

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

equation

equation (51)

We prove there exist infinitely many primes P1, P2, P3 and P4 such that P5,…,Pk are primes.

We have best asymptotic formula:

equation

equation (52)

Acknowledgements

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

References

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