The New Prime theorems（591）-（640）
Jiang, Chun-Xuan (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 Jiang function we are able to prove almost all prime problems in prime distribution. This is the Book proof. No mathematicians study prime problems and prove Riemann hypothesis. In this paper using Jiang function J2( )
we prove that the new prime theorems (591)-（640) contain infinitely many prime solutions and no prime solutions. From (6) we are able to find the smallest solution k(N0, 2) 1
. This is the Book theorem.
这开创新的素数理论新时代, 将产生一大批数学家。
[Jiang, Chun-Xuan (蒋春暄). The New Prime theorems（591）-（640）. Academ Arena 2016;8(1s): 354-408 (ISSN 1553-992X). http://www.sciencepub.net/academia. 8. doi:10.7537/marsaaj0801s1608.
Keywords: new; prime theorem; Jiang Chunxuan
Analytic and combinatorial number theory (August 29-September 3, ICM2010) is a conjecture. The sieve methods and circle method are outdated methods which cannot prove twin prime conjecture and Goldbach’s conjecture. The papers of Goldston-Pintz-Yildirim and Green-Tao are based on the Hardy-Littlewood prime k-tuple conjecture (1923). But the Hardy-Littlewood prime k-tuple conjecture is false:
(http://www.wbabin.net/math/xuan77.pdf)
(http://vixra.org/pdf/1003.0234v1.pdf). 我们不知 Goldston-Pintz-Yildirim-Green-Tao 他们在研究什么？很 长很长论文和素数没有任何联系, 没有任何有用结果, 只有 IAS 支持他们, 发表他们论文. 这就是当代最高 水平。
The world mathematicians read Jiang’s book and papers. In 1998 Jiang disproved Riemann hypothesis. In 1996 Jiang proved Goldbach conjecture and twin prime conjecture. Using a new analytical tool Jiang invented: the Jiang function, Jiang prove almost all prime problems in prime distribution. Jiang established the foundations of Santilli’s isonumber theory. China rejected to speak the Jiang epoch-making works in ICM2002 which was a failure congress.
China considers Jiang epoch-making works to be pseudoscience. Jiang negated ICM2006 Fields medal (Green and Tao theorem is false) to see.
(http://www.wbabin.net/math/xuan39e.pdf) (http://www.vixra.org/pdf/0904.0001v1.pdf).
There are no Jiang’s epoch-making works in ICM2010. It cannot represent the modern mathematical level.
Therefore ICM2010 is failure congress. China rejects to review Jiang’s epoch-making works. IMU should support Jiang epoch-making prime theory and the Book theorem to see[ new prime k-tuple theorems (1)-(20)] and [the new prime theorems (1)-(590)]: (http://www.wbabin.net/xuan.htm#chun-xuan) (http://vixra.org/numth/).
The New Prime theorem（591）
, 1102 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1102 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1102 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1102
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 that is for any k there are infinitely many primes P such that each of
jp1102
+k j
is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1
. Substituting it into (2) 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]
1102
2 1 1( , 2) : ~ ( )
(1102) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3, 59,1103
. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3, 59,1103,
(1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3,59,1103
. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3, 59,1103，
(1) contain infinitely many prime solutions
The New Prime theorem（592）
, 1104 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1104 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1104 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1104
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 that is for any k there are infinitely many primes P such that each of
jp1104
+k j
is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1
. Substituting it into (2) 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]
1104
2 1 1( , 2) : ~ ( )
(1104) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3,5, 7,13,17, 47,139, 277
. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3,5, 7,13,17, 47,139, 277,
(1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3,5, 7,13,17, 47,139, 277
. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3,5, 7,13,17, 47,139, 277，
(1) contain infinitely many prime solutions
The New Prime theorem（593）
, 1106 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1106 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1106 ( 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( ) 2[ 1 ( )]
J P P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1104
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 that is for any k there are infinitely many primes P such that each of
jp1106+k j is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1. Substituting it into (2) 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]
1106
2 1 1( , 2) : ~ ( )
(1106) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3,
(1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3，
(1) contain infinitely many prime solutions
The New Prime theorem（594）
, 1108 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Abstract
Using Jiang function we prove that
jP1108 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1108 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1108
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 that is for any k there are infinitely many primes P such that each of
jp1108
+k j
is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1
. Substituting it into (2) 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]
1108
2 1 1( , 2) : ~ ( )
(1108) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3, 5,1109
. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3, 5,1109,
(1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3,5,1109
. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3,5,1109，
(1) contain infinitely many prime solutions
The New Prime theorem（595）
, 1110 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1110 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1110 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1110
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 that is for any k there are infinitely many primes P such that each of
jp1110+k j is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1. Substituting it into (2) 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]
1110
2 1 1( , 2) : ~ ( )
(1110) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3, 7,11,31, 223. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3, 7,11,31, 223,
(1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3, 7,11,31, 223.
From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3, 7,11,31, 223，
(1) contain infinitely many prime solutions
The New Prime theorem（596）
, 1112 ( 1, , 1) P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1112 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1112 ( 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( ) 2[ 1 ( )]
J P P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1112
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 that is for any k there are infinitely many primes P such that each of
jp1112
+k j
is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1
. Substituting it into (2) 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]
1112
2 1 1( , 2) : ~ ( )
(1112) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3, 5,557. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3, 5,557
,
(1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3,5,557.
From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3,5,557，
(1) contain infinitely many prime solutions
The New Prime theorem（597）
, 1114 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1114 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1114 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1114
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 that is for any k there are infinitely many primes P such that each of
jp1114+k j is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1. Substituting it into (2) 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]
1114
2 1 1( , 2) : ~ ( )
(1114) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3,
(1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3，
(1) contain infinitely many prime solutions
The New Prime theorem（598）
, 1116 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1116 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1116 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1116
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 that is for any k there are infinitely many primes P such that each of
jp1116+k j is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1. Substituting it into (2) 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]
1116
2 1 1( , 2) : ~ ( )
(1116) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3, 5, 7,13,373,1117. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3, 5, 7,13,373,1117
, (1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3,5, 7,13,373,1117.
From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3,5, 7,13,373,1117，
(1) contain infinitely many prime solutions
The New Prime theorem（599）
, 1118 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1118 k j contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1118 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P
is the number of solutions of congruence
1 1118
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 that is for any k there are infinitely many primes P such that each of
jp1118
+k j
is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1
. Substituting it into (2) 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]
1118
2 1 1( , 2) : ~ ( )
(1118) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3,
(1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3. From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3，
(1) contain infinitely many prime solutions
The New Prime theorem（600）
, 1120 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1120 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1120 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1120
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 that is for any k there are infinitely many primes P such that each of
jp1120
+k j
is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1
. Substituting it into (2) 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]
1120
2 1 1( , 2) : ~ ( )
(1120) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3,5,11,17, 29, 41, 71,113, 281
. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3,5,11,17, 29, 41, 71,113, 281
, (1) contain no prime solutions. 1 is not a prime.
Example 2. Let k3,5,11,17, 29, 41, 71,113, 281.
From (2) and (3) we have
2( ) 0
J
（8）
We prove that for k3,5,11,17, 29, 41, 71,113, 281，
(1) contain infinitely many prime solutions
The New Prime theorem（601）
, 1122 ( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
[email protected] Abstract
Using Jiang function we prove that
jP1122 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
, 1122 ( 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( ) 2[ 1 ( )]
P
J P P
（2）
where P P
，( )P is the number of solutions of congruence
1 1122
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 that is for any k there are infinitely many primes P such that each of
jp1122
+k j
is a prime.
Using Fermat’s little theorem from (3) we have ( )P P1
. Substituting it into (2) 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]
1122
2 1 1( , 2) : ~ ( )
(1122) ( ) log
k
k k k k
J N
N P N jP k j prime
N
（6）
where ( ) ( 1)
P P
.
From (6) we are able to find the smallest solution k(N0, 2) 1
. Example 1. Let k 3, 7, 23, 67,103,1123
. From (2) and(3) we have
2( ) 0
J
（7）
we prove that for k 3, 7, 23, 67,103,1123,
(1) contain no prime solutions. 1 is not a prime.
