The New Prime theorems(241)-(290)
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)
jiangchunxuan@sohu.com, cxjiang@mail.bcf.net.cn, jcxuan@sina.com, Jiangchunxuan@vip.sohu.com, jcxxxx@163.com
Abstract: Using Jiang function
J
2( )
we prove that the new prime theorems (241)-(290) contain infinitely many prime solutions and no prime solutions.
[Jiang, Chun-Xuan (蒋春暄). The New Prime theorems(241)-(290). Academ Arena 2016;8(1s): 1-46]. (ISSN 1553-992X). http://www.sciencepub.net/academia. 1. doi:10.7537/marsaaj0801s1601.
Keywords: new; prime; theorem; Jiang Chunxuan; mathematics; science; number; function The New Prime theorem(241)
,
402( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
402 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
402( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 402
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
402
2 1 1( , 2) : ~ ( )
(402) ( ) 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
. From (2) and(3) we have
2
( ) 0
J
(7)
we prove that for
k 3, 7
, (1) contain no prime solutions Example 2. Let
k 3, 7
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3, 7
(1) contain infinitely many prime solutionsThe New Prime theorem(242)
,
404( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
404 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
404( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 404
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
404
2 1 1( , 2) : ~ ( )
(404) ( ) 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
. From (2) and(3) we have2
( ) 0
J
(7)
We prove that for
k 3,5
(1) contain no prime solutions.Example 2. Let k 5. From (2) and (3) we have
2
( ) 0 J
(8)
We prove that for k5, (1) contain infinitely many prime solutions
The New Prime theorem(243)
,
406( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
406 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
406( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 406
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
406
2 1 1( , 2) : ~ ( )
(406) ( ) 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 3,59
. From (2) and(3) we have2
( ) 0
J
(7)
We prove that for
k 3,59
, (1) contain no prime solutions.
Example 2. Let
k 3,59
. From (2) and (3) we have2
( ) 0
J
(8)
We prove that for
k 3,59
(1) contain infinitely many prime solutions
The New Prime theorem(244)
,
408( 1, , 1) P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
408 k j
contain infinitely many prime solutions and no prime solutions.Theorem. Let k be a given odd prime.
,
408( 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
PP P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 408
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
408
2 1 1( , 2) : ~ ( )
(408) ( ) 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, 7,13,103,137, 409
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3,5, 7,13,103,137, 409
(1) contain no prime solutions.Example 2. Let
k 3,5, 7,13,103,137, 409
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3,5, 7,13,103,137, 409
(1) contain infinitely many prime solutionsThe New Prime theorem(245)
,
410( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
410 k j
contain infinitely many prime solutions and no prime solutions.Theorem. Let k be a given odd prime.
,
410( 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
PP P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 410
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
410
2 1 1( , 2) : ~ ( )
(410) ( ) 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 3,11,83
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3,11,83
, (1) contain no prime solutions.Example 2. Let
k 3,11,83
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3,11,83
, (1) contain infinitely many prime solutionsThe New Prime theorem(246)
,
412( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
412 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
412( 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
PP P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 412
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
412
2 1 1( , 2) : ~ ( )
(412) ( ) 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
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3,5
, (1) contain no prime solutions.Example 2. Let k 5. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for k 5, (1) contain infinitely many prime solutions
The New Prime theorem(247)
,
414( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
414 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
414( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 414
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
414
2 1 1( , 2) : ~ ( )
(414) ( ) 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, 7,19,97,139
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3, 7,19,97,139
, (1) contain no prime solutions.Example 2. Let
k 3, 7,19, 97,139
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3, 7,19, 97,139
, (1) contain infinitely many prime solutionsThe New Prime theorem(248)
,
416( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
416 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
416( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 416
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
416
2 1 1( , 2) : ~ ( )
(416) ( ) 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, 53
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3,5,17, 53
, (1) contain no prime solutions.
Example 2. Let
k 3,5,17,53
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3,5,17,53
, (1) contain infinitely many prime solutionsThe New Prime theorem(249)
,
418( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
418 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
418( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 418
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
418
2 1 1( , 2) : ~ ( )
(418) ( ) 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, 419
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3, 23, 419
, (1) contain no prime solutions.Example 2. Let
k 3, 23, 419
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3, 23, 419
, (1) contain infinitely many prime solutionsThe New Prime theorem(250)
,
420( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
420 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
420( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 420
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
1
420 2
1
( , 2) : ~ ( )
(420) ( ) 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, 7,11,13, 29, 31, 43, 61, 71, 211, 421
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3,5, 7,11,13, 29, 31, 43, 61, 71, 211, 421
, (1) contain no prime solutions.Example 2. Let
k 3,5, 7,11,13, 29,31, 43, 61, 71, 211, 421
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3,5, 7,11,13, 29,31, 43, 61, 71, 211, 421
, (1) contain infinitely many prime solutionsThe New Prime theorem(251)
,
422( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
422 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
422( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 422
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
422
2 1 1( , 2) : ~ ( )
(422) ( ) 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 k 3, (1) contain no prime solutions 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(252)
,
424( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
424 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
424( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 424
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
424
2 1 1( , 2) : ~ ( )
(424) ( ) 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, 5,107
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3, 5,107
(1) contain no prime solutions.Example 2. Let
k 3,5,107
. From (2) and (3) we have2
( ) 0
J
(8)
We prove that for
k 3,5,107
, (1) contain infinitely many prime solutionsThe New Prime theorem(253)
,
426( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
426 k j
contain infinitely many prime solutions and no prime solutions.Theorem. Let k be a given odd prime.
,
426( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 426
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
426
2 1 1( , 2) : ~ ( )
(426) ( ) 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
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3, 7
, (1) contain no prime solutions.Example 2. Let
k 3, 7
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3, 7
(1) contain infinitely many prime solutionsThe New Prime theorem(254)
,
428( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com
Abstract
Using Jiang function we prove that
jP
428 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
428( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 428
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
428
2 1 1( , 2) : ~ ( )
(428) ( ) 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
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3,5
(1) contain no prime solutions.
Example 2. Let
k 3,5
. From (2) and (3) we have2
( ) 0
J
(8)
We prove that for
k 3,5
(1) contain infinitely many prime solutions
The New Prime theorem(255)
,
430( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
430 k j
contain infinitely many prime solutions and no prime solutions.Theorem. Let k be a given odd prime.
,
430( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 430
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
430
2 1 1( , 2) : ~ ( )
(430) ( ) 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 3,11, 431
. From (2) and(3) we have2
( ) 0
J
(7)
We prove that for
k 3,11, 431
, (1) contain no prime solutions.
Example 2. Let
k 3,11, 431
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3,11, 431
, (1) contain infinitely many prime solutions
The New Prime theorem(256)
,
432( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
432 k j
contain infinitely many prime solutions and no prime solutions.Theorem. Let k be a given odd prime.
,
432( 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
PP P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 432
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
432
2 1 1( , 2) : ~ ( )
(432) ( ) 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, 7,13,17,19,37, 73,109, 433
. From (2) and(3) we have2
( ) 0
J
(7)
We prove that for
k 3,5, 7,13,17,19,37, 73,109, 433
, (1) contain no prime solutions.Example 2. Let
k 3,5, 7,13,17,19, 37, 73,109, 433
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3,5,167
, (1) contain infinitely many prime solutions
The New Prime theorem(257)
,
434( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
434 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
434( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 434
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0 J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
434
2 1 1( , 2) : ~ ( )
(434) ( ) 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
The New Prime theorem(258)
,
436( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
436 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
436( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 436
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
436
2 1 1( , 2) : ~ ( )
(436) ( ) 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
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3,5
, (1) contain no prime solutions.Example 2. Let k5. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for k5, (1) contain infinitely many prime solutions
The New Prime theorem(259)
,
438( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
438 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
438( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 438
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
438
2 1 1( , 2) : ~ ( )
(438) ( ) 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, 439
. From (2) and(3) we have
2
( ) 0
J
(7)
We prove that for
k 3, 7, 439
, (1) contain no prime solutions.
Example 2. Let
k 3, 7, 439
. From (2) and (3) we have
2
( ) 0
J
(8)
We prove that for
k 3, 7, 439
, (1) contain infinitely many prime solutionsThe New Prime theorem(260)
,
440( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
440 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
440( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 440
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
440
2 1 1( , 2) : ~ ( )
(440) ( ) 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, 23, 41,89
. From (2) and(3) we have2
( ) 0
J
(7)
We prove that for
k 3,5,11, 23, 41,89
, (1) contain no prime solutions.Example 2. Let
k 3,5,11, 23, 41,89
. From (2) and (3) we have2
( ) 0 J
(8)
We prove that for
k 3,5,11, 23, 41,89
, (1) contain infinitely many prime solutionsThe New Prime theorem(261)
,
442( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
442 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
442( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence1 442
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
442
2 1 1( , 2) : ~ ( )
(442) ( ) 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, 443
. From (2) and(3) we have
2
( ) 0
J
(7)
we prove that for
k 3, 443
, (1) contain no prime solutions Example 2. Letk 3, 443
. From (2) and (3) we have2
( ) 0
J
(8)
We prove that for
k 3, 443
(1) contain infinitely many prime solutions
The New Prime theorem(262)
,
444( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract
Using Jiang function we prove that
jP
444 k j
contain infinitely many prime solutions and no prime solutions.
Theorem. Let k be a given odd prime.
,
444( 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 ( )]
P
J P P
(2)
where
PP
,
( ) P
is the number of solutions of congruence
1 444
1
0 (mod ), 1, , 1
k
j
jq k j P q P
(3)
If
( ) P P 2
then from (2) and (3) we have
2
( ) 0
J
(4)
We prove that (1) contain infinitely many prime solutions.
If
( ) P P 1
then from (2) and (3) we have
2
( ) 0
J
(5)
We prove that (1) contain no prime solutions [1,2]
If
J
2( ) 0
then we have asymptotic formula [1,2]
444
2 1 1( , 2) : ~ ( )
(444) ( ) 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, 7,13,149
. From (2) and(3) we have2
( ) 0
J
(7)
We prove that for
k 3,5, 7,13,149
(1) contain no prime solutions.
Example 2. Let
k 3,5, 7,13,149
. From (2) and (3) we have2
( ) 0
J
(8)
We prove that for
k 3,5, 7,13,149
, (1) contain infinitely many prime solutions
The New Prime theorem(263)
,
446( 1, , 1)
P jP k j j k
Chun-Xuan Jiang
Jiangchunxuan@vip.sohu.com Abstract