Academia Arena 2015;7(1s) http://www.sciencepub.net/academia
47
The New Prime theorem(28)
P 1 P m
and P 1 2 P n
Chun-Xuan Jiang
P.O.Box3924, Beijing100854, P.R. China. Jiangchunxuan@vip.sohu.com
Abstract: Using Jiang function we prove that P 1 P m
and P 1 2 P n
have infinitely many prime solutions.
[Chun-Xuan Jiang. The New Prime theorem(28) P 1 P m
and P 1 2 P n
. Academ Arena 2015;7(1s):
47-48]. (ISSN 1553-992X). http://www.sciencepub.net/academia. 28 Keywords: prime; theorem; function; number; new
Theorem 1. Let m be an even number.
P 1 P m
(1)
has infinitely many prime solutions.
Proof. We have Jiang function [1,2]
2 ( ) [ 1 ( )]
J P P P
, ( 2 ) where P P
, ( ) P is the number of solutions of congruence 0 (mod ), 1, , 1
q m P q P . ( 3 )
If P m
then ( ) P 0. ( ) P 1 othewise.
Substituting it into (2) we have.
2 3
( ) ( 2) 1 0
2
P P m
J P P
P
. ( 4 ) We prove that (1) has infinitely many primes solutions.
We have asymptotic formula [1,2]
2
2 1 2 2 3 2 2
( ) 1 1
( , 2) : ~ 2 1
( ) log P ( 1) P m 2 log
J N P N
N P N P prime
N P P N
. (5)
Where ( ) ( 1)
P P
.
In the same way we are able to prove that P 1 P m
has infinitely many prime solutions.
Theorem 2. Let n be an odd number.
1 2
P P n
has infinitely many prime solutions.
Proof. We have Jiang function [1,2]
2 ( ) [ 1 ( )]
P
J P P
, (6)
where ( ) P is the number of solutions of congruence
2 q n 0 (mod P ) ( 7 )
1, , 1 q P . If P n
then ( ) P 0. ( ) P 1 otherwise.
Substituting it into (6) we have
Academia Arena 2015;7(1s) http://www.sciencepub.net/academia
48
2 3
( ) ( 2) 1 0
2
P P n
J P P
P
. (8)
We prove that (6) has infinitely many prime solutions.
We have asymptotic formula [1,2]
2
2 1 2 2
( , 2) : ~ ( )
( ) log
J N
N P N P prime
N
. (9)
Remark. The prime number theory is basically to count the Jiang function J n 1 ( )
and Jiang prime k -tuple
singular series
1
2
( ) 1 ( ) 1
( ) 1 (1 )
( )
k
k
k P
J P
J P P
[1,2], which can count the number of prime number. The prime distribution is not random. But Hardy prime k -tuple singular series
( ) 1
( ) 1 (1 )
kP