k JPPkP ()(1)[(1)(2)]0  kP  Pk 

Academia Arena 2016;8(2) http://www.sciencepub.net/academia

151

New prime K-tuple theorems(1)

1

,

2

,

1 2

( 1, , )

P P P  jP  j j   k

and

P P P

¹

,

²

,

¹

 jP

²

 j j (  1,  , ) k

Chun-Xuan, Jiang (蒋春暄)

Institute for Basic Research, Palm Harbor, FL34682-1577, USA

And: P. O. Box 3924, Beijing 100854, China (蒋春暄，北京3924信箱，100854)

[email protected], [email protected], [email protected], [email protected], [email protected]

Abstract: Using Jinag funciton we prove that there exist infinitely many primes

P

¹

and

P

²

such that each of

1 2

P  jP  j

is prime and there exist infinitely many primes

P

¹

and

P

²

such that each of

P

¹

 jP

²

 j

is prime.

[Chun-Xuan, Jiang. New prime K -tuple theorems(1)

P P P

¹

,

²

,

¹

 jP

²

 j j (  1,  , ) k

and

1

,

2

,

1 2

( 1, , )

P P P  jP  j j   k

. Academ Arena 2016;8(2):151-152]. ISSN 1553-992X (print); ISSN 2158-771X (online). http://www.sciencepub.net/academia. 6. doi:10.7537/marsaaj08021606.

Keywords: new; prime; function; number

Theorem 1.

1

,

2

,

1 2

( 1, , )

P P P  jP  j j   k

（1） There exist infinitely many primes

P

¹

and

P

²

such that each of

P

¹

 jP

²

 j

is prime.

Proof. We have Jiang function [1, 2]

2

3

( ) [( 1) ( )]

P

J



  P 



P

（2）

where

  

^P

P

，

( )

P



is the number of solutions of congruence

1 2

1

( ) 0 (mod )

k

j

q jq j P





  

,

q

_i

 1,  , P  1, i  1, 2.

（3）

From (3) we have

If

k  P

_then

 ^{( )}

^P ^{k P}

⁽



²⁾

_{. If}

P  k

_then

 ^{( )}

^P 

⁽

^P

¹⁾⁽

^P

²⁾

_. From (3) and (2) we have

2

3

( )

3

( 1) [( 1) ( 2)] 0

P k

P k P

J  P P k P



 

       

（4）

For any positive integer

k

there exist inifinitely many primes

P

¹

and

P

²

such that each of

P

¹

 jP

²

 j

is prime.

We have asymptotic formula [1, 2]

 

2 3

1 1 2 1 2 2 2

( ,3) , : ~ ( )

( ) log

k

k k k

J N

N P P N P jP j prime

N

  

 



    

 

, （5）

where

( ) ( 1)

P

P

    

. Example 1.

1

,

2

,

1 2

1

P P P  P 

. （6）

Academia Arena 2016;8(2) http://www.sciencepub.net/academia

152 From (4) we have

2

3

( )

3

[( 1) ( 2)] 0

J 

P

P P

 



   

（7）

From (5) we have

2

2 3 3 3

( , 3) ~ 2 1 1

( 1) log

P

N N

P N





 

   

  

_（8）

Example 2.

1

,

2

,

1 2

1,

1

2

2

2

P P P  P  P  P 

. （9）

From (4) we have

2

3

( )

3

[( 1) 2( 2)] 0

J 

P

P P

 



   

（10）

From (5) we have

2 2

3

3 4 4

( ,3) ~ ( )

( ) log

J N

N N

  

 

. （11）

Example 3.

1

,

2

,

1 2

( 1, ,8)

P P P  jP  j j  

. （12）

From (4) we have

2

3

( ) 48

11

[( 1) 8( 2)] 0

J



P P P

     

. （13）

From (5) we have

8 2

3

9 10 10

( ,3) ~ ( )

( ) log

J N

N N

  

 

. （14）

Theorem 2.

1

,

2

,

1 2

( 1, , )

P P P  jP  j j   k

, （15）

we have Jiang function

2

3

( )

3

( 1) [( 1) ( 2)] 0

P k

P k P

J 

^

P P k P

 

       

. （16）

For any positive integer

k

there exist infinitely many primes

P

¹

and

P

²

such that each of

P

¹

 jP

²

 j

is prime.

we have asymptotic formula

 

2 3

1 1 2 1 2 2 2

( ,3) , : ~ ( )

( ) log

k

k k k

J N

N P P N P jP j prime

N

  

 



    

 

.（17） References

1. Chun-Xuan Jiang, Jiang’s function

J

_n¹

( ) 

in prime distribution. (http:// www. Wbabin. net/math/xuan2.

pdf) (http://vixra.org/pdf/0812.0004v2.pdf)

2. Chun-Xuan Jiang, The Hardy-Littlewood prime

k 

_tuple conjecture is false.

http://www.wbabin.net/math/xuan77.pdf

2/24/2016