The Simplest Proofs of Both Arbitrarily Long
Arithmetic Progressions of primes
Chun-Xuan Jiang
P. O. Box 3924, Beijing 100854
P. R. China
Jiangchunxuan@sohu.com
Abstract
Using Jiang functions J 2 ( ) , J 3 ( ) and J 4 ( ) we prove both arbitrarily long arithmetic
progressions of primes: (1) Pi 1  P1  di, ( P1 , d )  1 , i  1, 2,
n
the same Jiang function; (2) Pi 1  P1  g i, i  1, 2,
, k  1, n  1 ， g   P and
n
generalized
i  1,
arithmetic
progressions
of
, k  1, n  1 , which have
2 P  Pg
Pi  P  i g
primes
and
Pk i  Pn  ig ,
, k, n  2 .
The Green-Tao theorem is false, because they do not prove the twin primes theorem and
arithmetic progressions of primes [3].
In prime numbers theory there are both well-known conjectures that there exist both arbitrarily
long arithmetic progressions of primes. In this paper using Jiang functions J 2 ( ) , J 3 ( ) and
J 4 ( ) we obtain the simplest proofs of both arbitrarily long arithmetic progressions of primes.
Theorem 1. We define arithmetic progressions of primes:
P1 , P2  P1 d, P3  P12 ,d k, P
P ( k 1 ) d, (1 P, .d)
1
1
(1)
We rewrite (1)
P3  2 P2  P,1
Pj  ( j  1) P2  ( j  2) P1 , 3  j  k .
（2）
We have Jiang function [1]
J 3 ( )   ( P  1)2  X ( P)  ,
3 P
（3）
X ( P) denotes the number of solutions for the following congruence
k
  ( j  1)q2  ( j  2)q1   0(mod P),
（4）
j 3
where
q1  1, 2,
, P  1; q2  1, 2,
, P  1.
From (4) we have
J 3 ( )  (P  1 )  P(  1P) ( k  1 
) as   .
3 P  k
（5）
k P
We prove that there exist infinitely many primes P1 and P2 such that P3 ,
, Pk are all primes
for all k  3 . It is a generalization of Euclid and Euler proofs for the existence of infinitely many
primes [1].
We have the best asymptotic formula [1]
 k 1 ( N , 3 )  j ( P12) j ( P 12 )

where
p r i mj e ,k 3P 1P 2 N,  ,
J 3 ( ) k  2 N 2
(1  o(1)),
2 k ( ) log k N
（6）
   P,  ( )   ( P  1) ,
2 P
（7）
2 P
 is called primorials,  ( ) Euler function.
(6) is a generalization of the prime number theorem  ( N ) 
N
(1  o(1)) [1].
log N
Substituting (5) and (7) into (6) we have the best asymptotic formula
 k 1 ( N ,3) 
1
P k 2
P k 2 ( P  k  1) N 2


(1  o(1)) .
2 2 Pk ( P  1)k 1 k  P ( P  1)k 1 log k N
From (8) we are able to find the smallest solution
（8）
 k 1 ( N0 ,3)  1 for large k .
Grosswald and Zagier obtain heuristically even asymptotic formulae [2]. Let k  2 and d  2 .
From (1) we have twin primes theorem: P2  P1  2 . The Green-Tao theorem is false, because
they do not prove the twin primes theorem and arithmetic progressions of primes [3].
Example 1. Let k  3 . From (2) we have
P3  2P2  P1 .
（9）
From (5) we have
J 3 ( )   ( P  1)( P  2)   as   .
3 P
（10）
We prove that there exist infinitely many primes P1 and P2 such that P3 are primes. From (8)
we have the best asymptotic formula

 N2
1
N2
 2 ( N ,3)   1 
(1  o(1))  0.66016 3 (1  o(1)) .（11）

3 P
( P  1)2 log3 N
log N


Example 2. Let k  4 . From (2) we have
P3  2P2  P1 ,
P4  3P2  2P1.
（12）
From (5) we have
J 3 ( )  2  ( P  1)( P  3)   as   .
5 P
（13）
We prove that there exist infinitely many primes P1 and P2 such that P3 and P4 are all
primes. From (8) we have the best asymptotic formula
 3 ( N ,3) 
9 P 2 ( P  3) N 2

(1  o(1)) .
4 5 P ( P  1)3 log 4 N
（14）
Example 3. Let k  5 . From (2) we have
P3  2P2  P1 ,
P4  3P2  2P1 ,
P5  4P2  3P1 .
（15）
From (5) we have
J 3 ( )  2  ( P  1)( P  4)   as   .
5 P
（16）
We prove that there exist infinitely many primes P1 and P2 such that P3 , P4 and P5 are
all primes. From (8) we have the best asymptotic formula
27 P3 ( P  4) N 2
 4 ( N ,3) 

(1  o(1)) .
4 5 P ( P  1)4 log5 N
（17）
Theorem 2. From (1) we obtain
P4  P3  P2  P,1
Pj  P3  ( j  3) P2  ( j  3) P1 , 4  j  k .
（18）
We have Jiang function [1]
J 4 ( )   (( P  1)3  X ( P)),
3 P
（19）
X ( P) denotes the number of solutions for the following congruence
k
 (q3  ( j  3)q2  ( j  3)q1 )  0(mod P),
j 4
where
qi  1, 2,
From (20) we have
, P  1, i  1, 2,3.
（20）
J 4 ( ) 

3 P  ( k 1)
( P  1) 2  ( P  1) ( P  1) 2  ( P  2)( k  3)   
( k 1)  P
as   .
(21)
We prove there exist infinitely many primes P1 , P2 and P3 such that P4 ,
, Pk are all
primes for all k  4 .
We have the best asymptotic formula [1]
 k  2 ( N , 4)   P3  ( j  3) P2  ( j  3) P1  prime, 4  j  k , P1 , P2 , P3  N 

J 4 ( ) k 3 N 3
(1  o(1)).
6 k ( ) log k N
（22）
Substituting (7) and (21) into (22) we have
 k 2 ( N , 4)

(23)
1
P k 3
P k 3[( P  1) 2  ( P  2)( k  3)] N 3


(1  o(1)).
k

2
k

1
k
6 2 P( k 1) ( P  1) ( k 1) P
( P  1)
log N
From (23) we are able to find the smallest solution
 k 2 ( N0 , 4)  1 for large k .
Example 4. Let k  4 . From (18) we have
P4  P3  P2  P1
（24）
From (21) we have
J 4 ()   ( P  1)( P2  3P  3)   as   .
3 P
（25）
We prove there exist infinitely many primes P1 , P2 and P3 such that P4 are primes.From (23)
we have
1

1
 N3
 2 ( N , 4)   1 
(1  o(1)).

3 3 P
( P  1)3 log 4 N


（26）
From (1) We obtain the following equations:
 k 3 ( N ,5)   P4  ( j  3) P3  ( j  2) P2  P1  prime,5  j  k , P1 ,
1 J 5 ( ) k  4 N 4

(1  o(1))
24  k ( ) log k N
(27)
 k  4 ( N , 6)   P5  ( j  4) P4  ( j  4) P3  P2  P1  prime, 6  j  k , P1 ,

, P4  N 
1 J 6 ( ) k 5 N 5
(1  o(1))
120  k ( ) log k N
, P5  N 
Theorem
(28)
3. We define arithmetic progressions of primes:
Pi 1  P12  di, i  1, 2,
, k 1 .
（29）
From (29) we have
P3  2P2  P12 , Pj  ( j 1) P2  ( j  2) P12 , 3  j  k .
(30)
We have Jiang function [1]
J 3 ( )   ( P  1)2  X ( P)  ,
3 P
(31)
X ( P) denotes the number of solutions for the following congruence
k
 ( j  1)q2  ( j  2)q12   0(mod P),
(32)
j 3
where q1  1, 2,
, P 1 ,
q2  1, 2,
, P 1.
From (32) we have
J 3 ( )   ( P  1)  ( P  1)( P  k  1)   as   .
3 P  k
（33）
k P
We prove that there exist infinitely many primes P1 and P2 such that P3 ,
, Pk are all primes
for all k  3 . We have the best asymptotic formula [1]
 k 1 ( N ,3)  ( j  1) P2  ( j  2) P12  prime,3  j  k , P1 , P2  N 
1 J 3 ( ) k 2 N 2
 k 1
(1  o(1)).
2
 k ( ) log k N
(34)
Substituting (7) and (33) into (34) we have
 k 1 ( N ,3) 
1
P k 2
P k 2 ( P  k  1) N 2


(1  o(1)) .
2k 1 2 Pk ( P  1)k 1 k  P ( P  1)k 1 log k N
(35)
Theorem 4. We define arithmetic progressions of primes:
Pi 1  P15  di, i  1, 2,
, k 1.
（36）
From (36) we have
P4  P3  P2  P15 , Pj  P3  ( j  3) P2  ( j  3) P15 , 4  j  k .
(37)
We have Jiang function [1]
J 4 ( ) 

3 P  ( k 1)
( P  1) 2  ( P  1) ( P  1) 2  ( P  2)( k  3)   
( k 1)  P
as   .
(38)
We prove that there exist infinitely many primes P1 , P2 and P3 such that P4 ,
, Pk are all
primes for all k  4 ..
We have the best asymptotic formula
 k  2 ( N , 4 ) P 3 j ( P3 2) j ( P 531)

p r i mj e ,k 4P P1 P2 , 3N,  ,
1 J 4 ( ) k 3 N 3
(1  o(1)).
6  5k 3  k ( ) log k N
（39）
Theorem 5. We define arithmetic progressions of primes:
Pj 1  P1n  di, i  1, 2,
, k 1, n  1.
（40）
From (40) we have
P3  2P2  P1n , Pj  ( j 1) P2  ( j  2) P1n .
(41)
We have Jiang function [1]
J 3 ( )   ( P  1)  ( P  1)( P  k  1)  
3 P  k
k P
as
  .
We prove that there exist infinitely many primes P1 and P2 such that P3 ,
（42）
, Pk are all primes
for all k  3 .
We have the best asymptotic formula [1]
 k 1 ( N ,3)  ( j  1) P2  ( j  2) P1n  prime,3  j  k , P1 , P2  N 

（43）
J 3 ( ) k  2 N 2
1
.
2  n k  2  k ( ) log k N
Substituting (7) and (42) into (43) we have
1
P k 2
P k 2 ( P  k  1) N 2
 k 1 ( N ,3) 


(1  o(1)) .
2  nk 2 2 Pk ( P  1)k 1 k  P ( P  1)k 1 log k N
（44）
Theorem 6. We define arithmetic progressions of primes:
Pj 1  P1n  di, i  1, 2,
, k 1, n  1.
（45）
From (45) we have
P4  P3  P2  P1n , Pj  P3  ( j  3) P2  ( j  3) P1n , 4  j  k .
(46)
We have Jiang function [1]
J 4 ( ) 

3 P  ( k 1)
( P  1) 2  ( P  1) ( P  1) 2  ( P  2)( k  3)   
( k 1)  P
as   .
(47)
We prove that there exist infinitely many primes P1 , P2 and P3 such that P4 ,
, Pk are all
primes for all k  4 ..
We have the best asymptotic formula [1]
 k 2 ( N , 4)  P3  ( j  3) P2  ( j  3) P1n  prime, 4  j  k , P1 , P2 , P3  N 
1 J 4 ( ) k 3 N 3

(1  o(1)).
6  nk 3  k ( ) log k N
（48）
Substituting (7) and (47) into (48) we have
 k 2 ( N , 4)

1
6  n k 3
P k 3
P k 3[( P  1) 2  ( P  2)(k  3)] N 3

(1  o(1)).
2 P  ( k 1) ( P  1) k  2 ( k 1)  P
( P  1) k 1
log k N

(49)
Theorem 7. We define another arithmetic progressions of primes [1, 4]:
Pi 1  P1  g i, i  1, 2,
where
g  
2 P  Pg
, k 1
（50）
is called a common difference, Pg is called g  th prime.
We have Jiang function [1, 4]
J 2 ( )   ( P  1  X ( P)) ,
（51）
3 P
X ( P) denotes the number of solutions for the following congruence
k 1
 (q  g i)  0(mod P),
（52）
i 1
where
q  1, 2,
, P 1 .
If P  g , then X ( P )  0 ; X ( P)  k  1 otherwise. From (52) we have
J 2 ( )   ( P  1)  ( P  k ) .
3 P  Pg
（53）
Pg 1  P
If k  Pg 1 then J 2 ( Pg 1 )  0 , J 2 ( )  0 , there exist finite primes P1 such that P2 ,
, Pk
are all primes. If k  Pg 1  1 then J 2 ( )  0 there exist infinitely many primes P1 such that
P2 ,
, Pk are all primes.
Let k  Pg 1  1 . From (50) we have
Pi 1  P1  g i, i  1, 2,
, Pg 1  2 .
（54）
From (53) we have [1, 4]
J 2 ( )   ( P  1)  ( P  Pg 1  1)  
3 P  Pg
Pg 1  P
as
 
We prove that there exist infinitely many primes P1 such that P2 ,
（55）
, PPg 1 1 are all primes for all
Pg 1 .
We have the best asymptotic formula [1, 4]
P
g 1 1
( N , 2)  P1  g i  prime,1  i  Pg 1  2, P1  N 
2
J 2 ( ) g 1
N

(1  o(1))
Pg 1 1
P 1
( ( ))
(log N ) g 1
P
Substituting (7) and (55) into (56) we have
.
（56）
P
g 1 1
( N , 2) 
 P 
 

2 P  Pg P  1


Pg 1  2
 
P
Pg 1  2
( P  Pg 1  1)
( P  1)
Pg 1  P
Pg 1 1
From (57) we are able to find the smallest solutions
N
(1  o(1)).
P 1
(log N ) g 1
P
g 1 1
(57)
( N0 , 2)  1 for large  g .
Example 5. Let P1  2 , 1  2 , P2  3 . From (54) we have the twin primes theorem
P2  P1  2.
（58）
From (55) we have
J 3 ( )   ( P  2)   as   ,
（59）
3 P
We prove that there exist infinitely many primes P1 such that P2 are primes. From (57) we
have the best asymptotic formula

 2 ( N , 2)  2  1 
3 P

1  N
(1  o(1)) .

( P  1) 2  log 2 N
（60）
2  6 , P3  5 . From (54) we have
Example 6. Let P2  3 ,
Pi 1  P1  6i, i  1, 2,3.
（61）
From (55) we have
J 2 ( )  2  ( P  4)   as   .
（62）
5 P
We prove that there exist infinitely many primes P1 such that P2 , P3 and P4 are all primes.
From (57) we have the best asymptotic formula
P3 ( P  4) N
(1  o(1)) .
5 P ( P  1) 4 log 4 N
 4 ( N , 2)  27 
Example 7. Let P9  23 ,
（63）
9  223092870 , P10  29 . From (54) we have
Pi 1  P1  223092870i, i  1, 2,
, 27.
（64）
From (55) we have
J 2 ( )  36495360  ( P  28)   as  
29  P
We prove that there exist infinitely many primes P1 such that P2 ,
（65）
, P28 are all primes. From
(57) we have the best asymptotic formula
 P 
 28 ( N , 2)   

2  P  23 P  1


27
P 27 ( P  28) N
(1  o(1)) . （66）
29  P
( P  1) 28 log 28 N

From (66) we are able to find the smallest solutions
 28 ( N0 , 2)  1 .
Theorem 8. We define another arithmetic progressions of primes:
Pi 1  P1n  g i, i  1, 2,
, k 1, n  1.
（67）
We have Jiang function [1]
J 2 ( )   ( P  1  X ( P)) ,
3 P
（68）
X ( P) denotes the number of solutions for the following congruence
k 1
 (q1n  g i)  0(mod P),
i 1
where q1  1, 2,
（69）
P 1 .
If X ( P )  P  1 and J 2 ( P)  0 , then there exist finite primes P1 such that P2 ,
Pk are
primes. If X ( P )  P  1 and J 2 ( )  0 , then there exist infinitely many primes P1 such that
P2 ,
, Pk are all prime for all Pg .
We have the best asymptotic formula [1]
 k ( N , 2)  P1n  g i  prime,1  i  k  1, P1  N 

1 J 2 ( ) k 1 N
(1  o(1)).
n k 1  k ( ) log k N
(70)
Example 8. Let n  2 , k  3 and  g  6 . From (67) we have
P2  P12  6 ,
P3  P12  12 ,
P4  P12  18
（71）
We have Jiang function [1]

 6   3   2  
J 2 ( )  2   P  4             as  
5 P
 P   P   P 

（72）
 6   3 
 2 
 ,   and   denote the Legendre symbols.
 P   P
 P 
where 
We prove that there exist infinitely many primes P1 such that P2 , P3 and P4 are all primes.
We have the best asymptotic formula [1]
 4 ( N , 2)  P12  6i  prime, i  1, 2,3, P1  N 

1 J 2 ( ) 3 N
(1  o(1)).
8  4 ( ) log 4 N
(73)
We shall move on to the study of the generalized arithmetic progression of consecutive primes [5].
A generalized arithmetic progression of consecutive primes is defined to be the sequence of
primes,
, P  kg and Pn  g , Pn  2g ,
P, P  g , P  2g ,
, Pn  kg ,
where P is the first term, n  2 . For example, 5, 11, 17, 23, and 31, 37, 43, is a generalized
arithmetic progression of primes with P  5 ,  g  6 , k  3 and n  2 .
Theorem 9. We define the generalized arithmetic progressions:
Pi  P  i g and Pk i  Pn  ig
where i  1,
(74)
, k, n  2 .
We have Jiang function [1]
J 2 ( )    P  1  X ( p)  ,
3 P
（75）
X ( P) is the number of solutions of congruence
k
(q  ig )(q n  ig )  0(mod P),
i 1
q  1, 2,
（76）
P  1.
If X ( P )  P  1 and J 2 ( P)  0 , then there exist finite primes P such that P1 , P2 ,
, P2 k
are primes. If X ( P )  P  1 , J 2 ( )  0 , then there exist infinitely many primes P such that
P1 , P2 ,
, P2 k are all primes.
If J 2 ( )  0 , we have the best asymptotic formula of the number of primes P  N [1]
 2 k 1 ( N , 2) 
J 2 ( ) 2 k
N
(1  o(1)).
k 2 k 1
n  ( ) (log N )2 k 1
(77)
Example 9. Let  g  6, k  3 , and n  2 . From (74) we have
P1  P  6, P2  P  12, P3  P  18
and
P4  P 2  6, P5  P 2  12, P6  P 2  18 .
(78)
We have Jiang function [1]

 2   3   6  
J 2 ( )  12672   P  7            0
23 P
 P   P   P 

Since J 2 ( )   as
（79）
  , there exist infinitely many primes P such that P1 , , P6
are all primes.
From (77) we have
 7 ( N , 2) 
J 2 ( ) 6 N
(1  o(1)).
8 7 ( ) log 7 N
(80)
Remark. Theorems 1, 3 and 5 have the same Jiang function J 3 ( ) and theorems 2, 4 and 6 the
same Jiang function J 4 ( ) which have the same character. All irreducible prime equations have
the Jiang functions and the best asymptotic formulas [1]. In our theory there are no almost primes,
for example P1  P2 P3  2 and N  P1  P2 P3 are theorems of three genuine primes. Using
the sieve method, circle method, ergodic theory, harmonic analysis, discrete geometry, and
combinatories they are not able to attack twin primes conjecture, Goldbach conjecture, long
arithmetic progressions of primes and other problems of primes and to find the best asymptotic
formulas. The proofs of Szemerédic’s theorem are false, because they do not prove the twin
primes theorem and arithmetic progressions of primes [3, 6-9].
Acknowledgement the Author would like to thank Zuo Mao-Xian for helpful conversations.
References
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cryptograms, Fermat’s theorem and Goldbach’s conjecture, Inter. Acad. Press, 2002, MR
2004c: 11001, http: //www.i-b-r.org/docs/jiang/pdf
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(1982).
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Ann. Math.
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mathematics workshop, Fujian normal university, October 28 to 30, 1995.
[5] Chun-Xuan, Jiang, Generalized Arithmetic Progressions Pi  P  i g and
Pk i  Pm  ig , preprints, 2003.
[6] E. Szemerédi, On sets of integers containing no k elements in arithmetic progressions, Acta
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2006-10, in Beijing