 P d  3)105( xd )( d  2)15( xdxdxd  2)2()1()( P  )1()(  P d  3)105( xd )( d  2)15( xdxdxd  2)2()1()(

(1)

Academia Arena 2017;9(17s) http://www.sciencepub.net/academia

23

Gaps Among Products of m Primes

Jiang Chunxuan (蒋春暄)

Institute for Basic Research, Palm Harbor, FL 34682-1577, USA

And: P. O. Box 3924, Beijing 100854, China (蒋春暄，北京3924信箱，中国，100854) [email protected], [email protected], [email protected], [email protected],

[email protected], [email protected]

摘要 (Abstract): Theorem 1.

d(x)d(x1)d(x2)2 infinitely-often. （1）

where d(x)

represents the number of distinct prime factors of x_, d(x) 1,d(3)1

x

P , d(15)2

,

3 ) 105

( 

d .

Proof (see[1] p.146 theorem 3.1.154). Prime equations are

p₂ 10p₁1, p₃ 15p₁2, p₄ 6p₁1

（2） We have Jiang function

0 ) 4 ( 3 ) (

2  7  

 P

J

P



, （3）

where P

P







2



We prove that J₂()0

there exist infinitely many primes P¹

such that P²

, P³

, P⁴

are primes.

We have asymptotic formula

 

N N P J

P P

N P

N 4 4

2 1

1 1

1

4 ( ) log

)

~ ( 1 6 , 2 15 , 1 10 : )

2 ,

(  



      

, （4）

where ( ) ( 1)

2 



 P

P





.

From (2) we have 3p² 130p¹42p³

, 3p² 230p¹ 55p⁴

. We prove that there exist infinitely many triples of consecutive integers, each being the products of two distinct primes.

[Jiang Chunxuan (蒋春暄). Gaps Among Products of m Primes. Academ Arena 2017;9(17s): 23-26]. (ISSN 1553-992X). http://www.sciencepub.net/academia. 5. doi:10.7537/marsaaj0917s1705.

关键词 (Keywords): Theorem; number; factor; Prime; equation；function Theorem 1.

d(x)d(x1)d(x2)2

infinitely-often. （1） where d(x)

represents the number of distinct prime factors of x_, d(x) 1,d(3)1

x

P , d(15)2

,

3 ) 105

( 

d .

Proof (see[1] p.146 theorem 3.1.154). Prime equations are

p₂ 10p₁1, p₃ 15p₁2, p₄ 6p₁1

（2） We have Jiang function

0 ) 4 ( 3 ) (

2  7  

 P

J

P



, （3）

where P

P







2



We prove that J₂()0

there exist infinitely many primes P¹

such that P²

, P³

, P⁴

are primes.

(2)

24 We have asymptotic formula

 

N N P J

P P

N P

N 4 4

2 1

1 1

1

4 ( ) log

)

~ ( 1 6 , 2 15 , 1 10 : )

2 ,

(  



      

, （4）

where ( ) ( 1)

2 



 P

P





.

From (2) we have 3p² 130p¹42p³

, 3p² 230p¹ 55p⁴

. We prove that there exist infinitely many triples of consecutive integers, each being the products of two distinct primes.

Theorem 2.

1 )

2 ( ) 1 ( )

(x d x d x m

d infinitely-often （5）

Proof (see [1] p.148, theorem 3.1.158). Suppose that u,u1

and u2 are three consecutive integers, each being the products of m1 distinct primes. Let M u(u1)(u2). We define the three prime equations

2 1

1

2  P 

u P M

,

1 1 2

1

3 

  P u P M

,

2 1 2

1

4 

  P u P M

（6）

Using Jiang function J₂() we prove that there exist infinitely many primes P¹ such that P²_, P₃

and P⁴

are primes.

From (6) we have

uP₂ 2MP₁u

,

3 1

1

2 1 ( 1)

1 ) 2 1 ( 1 2

1 P u P

u u M u

MP

uP  



 



 

 









4 1

1

2 1 ( 2)

2 ) 2 2 ( 2 2

2 P u P

u u M u

MP

uP  



 



 

 









We prove

d(x)d(x1)d(x2)m1

infinitely-often. （7） Theorem 3.

2 ) 4 ( ) 2 ( )

(x d x d x 

d infinitely-often （8）

Proof [1,2,3]. Prime equations are

P₂ 70P₁ 1, P₃ 42P₁1, P₄ 30P₁1

（9）

and P⁴

are primes.

Frome (9) we have

3P² 210P¹ 3, 3P² 2210P¹55(42P¹1)5P³

3P² 4210P¹77(30P¹ 1)7P⁴

（10） We prove

2 ) 4 3 ( ) 2 3 ( ) 3

( P₂ d P₂  d P₂  

d infinitely-often. （11）

Theorem 4.

d(x)d(x2)d(x4)m1

infinitely-often. （12）

Proof [1, 2, 3]. Suppose that u,u2_and _u_₄ are three odd integers, each being the products of m1 distinct primes. Let M u(u2)(u4)

We define three prime equations

2 1

1

2  P 

u P M

,

2 1 2

1

3 

  P u P M

,

4 1 2

1

4 

  P u P M

（13）

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25

and P⁴

are primes.

From (13) we have uP₂ 2MP₁ u

,

2 2

2 ₁

2   MP u

uP ¹ 1 ( 2) ³

2 ) 2 2

( P u P

u

u M  



 



 

 



,



 



 

 







 1)

4 ) 2 4 ( 4

4 ₁ ₁

2 P

u u M u

MP uP

) 4

4 (u P

 . （14） We prove

1 )

4 ( ) 2 ( )

(x d x d x m

d infinitely-often. （15）

Theorem 5.

d(x)d(x1)d(x2)d(x4)m1

Proof. From (9) we have prime equations P₂ 70P₁1

, P₃ 105P₁ 2

, P₄ 42P₁ 1

, P₅ 30P₁1

（17） Using Jiang function we prove there exist infinitely many primes P¹

such that P²

, P³

, P⁴

and P⁵

are primes.

From (17) we have 3P₂ 210P₁3

3P² 1210P¹ 42(105P¹ 2)2P³

4 1

1

2 2 210 5 5(42 1) 5

3P   P   P   P

5 1

1

2 4 210 7 7(30 1) 7

3P   P   P   P

. （18） Using Jiang function we prove

d(x)d(x1)d(x2)d(x4)m1

1 )

10 ( ) 8 ( ) 4 ( ) 2 ( ) 1 ( )

(x d x d x d x d x d x m

d infinitely-often.

（20） Using Jiang function J₂()

we are able to prove d(x)d(xn)m1

infinitely-often. （21） d(x)d(x53)d(x73)d(xP3)m1 infinitely-often. （22）

Goldston et. al prove only d(x)d(xn6)2 infinitely-often [4].

我们发现素数分布新的规律，这个问题比哥德巴赫猜想要难一万倍。这是人类最伟大数学发现。欧拉高斯没接触这个问题,20世纪最伟大数学家埃尔德什开始关注这个问题, 但也没得出有用结果。最近国际顶尖数

学家Goldston等正在研究这个问题。得到国际数学界广泛的支持和关注，但文章都发表在著名杂志上，但没

有得出任何实质性进展，蒋春暄在2002年[1]就彻底证明了它，但国内外数学家都读了它，都不说话，看到文献[4]后，我们决定写本文，如不用Jiang函数，再过两百年也不一定能证明它，国内更无人研究它，这才是研究方向！2009年1月10日蒋春暄为休息去参加宋正海讲座在公共汽车上发现公式(22),1月11日发现定理五。这样算一篇完整论文。

References

1. Chun-Xuan Jiang, Foundations of Santilli’s isonumber theory with applicatio applications to new cryptograms, Fernat’s theorem and Goldbach’s conjecture. Inter. Acad. Press, 2002, MR2004c:11001, (http://www.i-b-r.org/docs/jiang.pdf) (http://www.wbabin.net/math/xuan13. pdf).

2. Chun-Xuan Jiang, on the consecutive integers ni1(i1)Pⁱ

, (http:// www. wbabin.net

(4)

26 /math/xuan40.pdf).

3. Chun-Xuan Jiang, Jiang’s funciton J_n_₁()

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

pdf).

4. D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between products of two primes, Proc.

London Math. Soc, (3) 98 (2009) 741-774.

5/7/2017