Academia Arena 2015;7(1s) http://www.sciencepub.net/academia
61
The New Prime theorem(34)
,
2( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
P. O. Box 3924, Beijing 100854, P. R. China jiangchunxuan@vip.sohu.com
Abstract: Using Jiang function we prove that if J
2( ) 0
then there are infinitely many primes P such that each of
jP
2 k j is a prime, if J
2( ) 0
then there are finitely many primes P such that each of
jP
2 k j is a prime.
[Chun-Xuan Jiang. The New Prime theorem(34)
,
2( 1, , 1)
P jP k j j k
. Academ Arena 2015;7(1s):
61-63]. (ISSN 1553-992X). http://www.sciencepub.net/academia. 34 Keywords: prime; theorem; function; number; new
Theorem . Let k be a given prime.
,
2( 1, , 1)
P jP k j j k
(
1
)Proof we have Jiang function [1,2]
2
( ) [ 1 ( )]
P
J P P
,
(2)where ( ) P is the number of solutions of congruence
1 2
1
0 (mod ), 1, , 1.
k
j
jq k j P q P
(3)
From (2) and (3) we have that if J
2( ) 0
then there are infinitely many primes P such that each of
jP
2 k j is a prime. If J
2( ) 0
then it has only finitely many prime solutions. If J
2( ) 0
we have asymptotic formula [1,2]
2
2 1 1( , 2) : ~ ( )
(2) ( ) log
k
k k k k
J N
N P N jP k j prime
N
(4)Exapmle 1. Let k 3 . From (1) we have.
2 2
, 2, 2 1
P P P
(5
)From (3) we have (3) 2 . From (2) we have
2
( ) 0
J
(
6
)We prove that (5) have only a solutions P 3 , P
2 2 11 , 2 P
2 1 19 .
Example 2. Let k 5 . From (1) we have
,
25 ( 1, 2, 3, 4)
P jP j j
(7
)P=7,j=1,53;j=2,101;j=3,149;j=4,197.
From (2) and (3) we have
2 7
1 6
( ) 4 [ 5 2( ) 2( )] 0
P
J P
P P
(
8
)We prove that there are infinitely many primes P such that each of
2
5
jP j is a prime.
Academia Arena 2015;7(1s) http://www.sciencepub.net/academia
62 From (4) we have
2
2 45 5 5
( , 2) : 5 ~ ( )
16 ( ) log
J N
N P N jP j prime
N
(9)
Example 3. Let k 7 . From (1) we have
,
27 ( 1, 2,3, 4,5, 6)
P jP j j
(10)From (2) and (3) we have
2 11
3 6 10
( ) 16 [ 7 2( ) 2( ) 2( )] 0
P
J P
P P P
(11)
We prove that there are infinitely many primes P such that each of
2
7
jP j is a prime.
From (4) we have
2
2 67 7 7
( , 2) : 7 ~ ( )
64 ( ) log
J N
N P N jP j prime
N
(
12
)Remark. The prime number theory is basically to count the Jiang function J
n1( )
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