Keywords: prime; theorem; function; number; new
The New Prime theorem（45）
, 10 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang [email protected]
Abstract: Using Jiang function we prove that
jP10 k j contain infinitely many prime solutions and no prime solutions.
Keywords: prime; theorem; function; number; new Theorem. Let k be a given odd prime.
, 10 ( 1, , 1)
P jP k j j k
. （1）
contain infinitely many prime solutions and no 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
1 10
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
2( ) 0 J
（8）
We prove that for k5, 7 and k11 (1) contain infinitely many prime solutions
The New Prime theorem（46）
, 12 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang [email protected]
Abstract: Using Jiang function we prove that
jP12 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 12 ( 1, , 1)
P jP k j j k
. （1）
contain infinitely many prime solutions and no 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
1 12
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
1
12 2
1
( , 2) : ~ ( )
(12) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Abstract: Using Jiang function we prove that
jP14 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 14 ( 1, , 1)
P jP k j j k
. （1）
contain infinitely many prime solutions and no 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
1 14
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
14
2 1 1( , 2) : ~ ( )
(14) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, (1) contain no prime solutions.
Example 2. Let k 3. From (2) and (3) we have
Abstract: Using Jiang function we prove that
jP16 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 16 ( 1, , 1)
P jP k j j k
. （1）
contain infinitely many prime solutions or no 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
1 16
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
1
16 2
1
( , 2) : ~ ( )
(16) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 5,17. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, 5,17 (1) contain no prime solutions.
Example 2. Let k 7,11,13 and k 17. From (2) and (3) we have
2( ) 0
J
（8）
We prove that fork7,11,13 and k17 (1) contain infinitely many prime solutions
Theorem. Let be a given odd prime.
, 18 ( 1, , 1)
P jP k j j k
. （1）
contain infinitely many prime solutions and no 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
1 18
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
18
2 1 1( , 2) : ~ ( )
(18) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 7,19. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, 7,19, (1) contain no prime solutions.
Example 2. Let k 5,11,13,17 and k19. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k5,11,13,17 and k 19, (1) contain infinitely many prime solutions
Abstract: Using Jiang function we prove that
jP20 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 20 ( 1, , 1)
P jP k j j k
. （1）
contain infinitely many prime solutions and no 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
1 20
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
20
2 1 1( , 2) : ~ ( )
(20) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 5,11. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, 5,11, (1) contain no prime solutions.
Example 2. Let k7 and k 11. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k7 and k11, (1) contain infinitely many prime solutions
Theorem. Let be a given odd prime.
, 22 ( 1, , 1)
P jP k j j k
, （1）
contain infinitely many prime solutions and no 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
1 22
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
22
2 1 1( , 2) : ~ ( )
(22) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 23. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k 3, 23, (1) contain no prime solutions.
Example 2. Let k 3, 23. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k 3, 23, (1) contain infinitely many prime solutions
Abstract
Using Jiang function we prove that
jP24 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 24 ( 1, , 1)
P jP k j j k , （1）
contain infinitely many prime solutions and no 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
1 24
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
24
2 1 1( , 2) : ~ ( )
(24) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 5, 7,13. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, 5, 7,13, (1) contain no prime solutions.
Example 2. Let k11 and k17. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k 11 and k17, (1) contain infinitely many prime solutions
Theorem. Let be a given odd prime.
, 26 ( 1, , 1)
P jP k j j k
, （1）
contain infinitely many prime solutions and no 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
1 26
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
26
2 1 1( , 2) : ~ ( )
(26) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, (1) contain no prime solutions.
Example 2. Let k3. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k 3, (1) contain infinitely many prime solutions
Abstract: Using Jiang function we prove that
jP28 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 28 ( 1, , 1)
P jP k j j k
, （1）
contain infinitely many prime solutions and no 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
1 28
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
28
2 1 1( , 2) : ~ ( )
(28) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 29. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, 29, (1) contain no prime solutions.
Example 2. Let k3, 29. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k 3, 29, (1) contain infinitely many prime solutions
Theorem. Let be a given odd prime.
, 30 ( 1, , 1)
P jP k j j k
. （1）
contain infinitely many prime solutions and no 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
1 30
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
30
2 1 1( , 2) : ~ ( )
(30) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 7,11, 31. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, 7,11, 31, (1) contain no prime solutions.
Example 2. Let k5,13,17,19, 23, 29 and k 31. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k 5,13,17,19, 23, 29 and k 31, (1) contain infinitely many prime solutions
Abstract: Using Jiang function we prove that
jP32 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 32 ( 1, , 1)
P jP k j j k
, （1）
contain infinitely many prime solutions and no 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
1 32
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
32
2 1 1( , 2) : ~ ( )
(32) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 5,17. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, 5,17, (1) contain no prime solutions.
Example 2. Let k7,11,13 and k17. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k 7,11,13 and k17, (1) contain infinitely many prime solutions
, 1 contain infinitely many prime solutions and no 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
1 34
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
34
2 1 1( , 2) : ~ ( )
(34) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, (1) contain no prime solutions.
Example 2. Let k 3. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3, (1) contain infinitely many prime solutions
Abstract: Using Jiang function we prove that
jP36 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 36 ( 1, , 1)
P jP k j j k
, （1）
contain infinitely many prime solutions and no 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
1 36
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
36
2 1 1( , 2) : ~ ( )
(36) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3, 5, 7,13,19, 37. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, 5, 7,13,19, 37, (1) contain no prime solutions.
Example 2. Let k 3, 5, 7,13,19, 37. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3, 5, 7,13,19, 37, (1) contain infinitely many prime solutions
Theorem. Let be a given odd prime.
, 38 ( 1, , 1)
P jP k j j k
, （1）
contain infinitely many prime solutions and no 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
1 38
1 0 (mod ), 1, , 1
k
j jq k j P q P
（3）
If ( )P P2 then from (2) and (3) we have
2( ) 0
J
（4）
We prove that (1) contain infinitely many prime solutions.
If ( )P P1 then from (2) and (3) we have
2( ) 0
J
（5）
We prove that (1) contain no prime solutions [1,2]
If J2( ) 0
then we have asymptotic formula [1,2]
38
2 1 1( , 2) : ~ ( )
(38) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
Example 1. Let k 3. From (2) and(3) we have
2( ) 0
J
（7）
We prove that for k3, (1) contain no prime solutions.
Example 2. Let k 3. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3, (1) contain infinitely many prime solutions
