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Gaps Among Products of m Primes

Chun-Xuan Jiang

P. O. Box 3924, Beijing 100854, P. R. China

Jiangchunxuan@vip.sohu.com

Abstract

Using Jiang function we prove gaps among products of prime:

infinitely-often,

where denotes the th prime.

Theorem 1. Let , gaps among products of two primes

infinitely-often. （1）

where represents the number of distinct prime factors of , , , .

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

（2）

We have Jiang function

, （3）

where

We prove that there exist infinitely many odd integers such that , and are primes.

We have asymptotic formula

, （4）

where .

From (2) we have

,

, （5）

From (5) we prove

infinitely-often. （6）

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

Theorem 2. Let , gaps among products of primes.

infinitely-often （7）

Proof (see [1] p.148, theorem 3.1.158). Suppose that and are three consecutive integers, each being the products of distinct primes. Let . We define the three prime equations

（8）

Using Jiang function we can prove that there exist infinitely many integers such that , and are primes.

From (8) we have

（9）

We prove

infinitely-often. （10）

Theorem 3. Let , gaps among products of two primes.

infinitely-often. （11）

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

（12）

Using Jiang function [1] we can prove that there exist infinitely many integers such that , and are primes.

Frome (12) we have

（13）

We prove

infinitely-often. （14）

Theorem 4. Let , gaps among products of primes.

infinitely-often. （15）

Proof [1, 2, 3]. Suppose that and are three odd integers, each being the products of distinct primes. Let

We define three prime equations

, , （16）

Using Jiang function [1] we can prove that there exist infinitely many integers such that , and are primes.

From (16) we have

,

,

. （17）

We prove

infinitely-often. （18）

Theorem 5. Let , gaps among products of primes.

infinitely-often. （19）

Proof. From (12) we have prime equations

, , , （20）

Using Jiang function [1] we can prove there exist infinitely many odd integers such that , , and are primes

From (20) we have

. （21）

We prove

infinitely-often. （22）

Using Jiang function we can prove that

infinitely-often. （23）

Theorem 6. Gaps among products of primes.

infintely-often. （24）

where denotes the th prime.

Proof. Let . We define the prime equations

, , , , .（25）

Using Jiang function [1] we can prove that there exist infinitely many odd integers such that , , are primes.

From (25) we have

,

,

,

,

……

. （26）

From（26）we have

infinitely-often.（27）

Using Jiang function [1] we can prove that

infinitely-often. （28）

Theorem 7. Gaps between products of two primes.

infinitely-often （29）

Proof. We define prime equations

. （30）

Using Jiang function we can prove that there exist infinitely many integers such that , and are primes.

From （30）we have

（31）

From （31）we prove

infinitely-often. （32）

Theorem 8. Gaps between products of primes.

infinitely-often （33）

Proof. Suppose that and are two integers, each being the products of distinct

primes.

Let . We define prime equations

（34）

Using Jiang function we can prove that there exist infinitely many integers such that , and are primes.

From （34）we have

（35）

From （35）we prove

infinitely-often. （36）

Theorem 9. Gaps between products of two primes.

infinitely-often （37）

Proof. We define prime equations

. （38）

Using Jiang function we can prove that there exist infinitely many integers such that , and are primes.

From （38）we have

（39）

From （39）we prove

infinitely-often. （40）

Theorem 10. Gaps between products of primes.

infinitely-often （41）

Proof. Suppose that and are two integers, each being the products of distinct

primes. Let . We define prime equations

（42）

Using Jiang function we can prove that there exist infinitely many integers such that , and are primes.

From （42）we have

（43）

From（43）we prove

infinitely-often. （44）

Theorem 11. Gaps between products of two primes.

infinitely-often （45）

Proof. Suppose . We define prime equations

. （46）

Using Jiang function we can prove that there exist infinitely many integers such that and are primes.

From （46）we have

（47）

From （47）we prove that

infinitely-often. （48）

Using Jiang function we can prove that

infinitely-often （49）

Theorem 12. Gaps between products of two primes.

We study general solutions of

infinitely-often （50）

Proof. We define a prime equation

. （51）

Using Jiang function we can prove that there exist infinitely many prime such that is a prime.

From （51）suppose . We define prime equations

（52）

Using Jiang function we can prove that there exist infinitely many integers such that and are primes.

From （52）we have

（53）

We prove that

infinitely-often. （54）

Theorem 13. Gaps between products of two primes.

infinitely-often （55）

Using we define prime equations,

（56）

From （56）we have

（57）

We redefine prime equations

（58）

From（58）we have

（59）

Theorem 14. Gaps between products of two primes.

infinitely-often （60）

Using we define prime equations,

（61）

From （61）we have

（62）

We redefine prime equations

（63）

From（63）we have

（64）

Theorem 15. Gaps between products of two primes.

infinitely-often （65）

Using we define prime equations,

（66）

From （66）we have

（67）

We redefine prime equations

（68）

From（68）we have

（69）

Theorem 16. Gaps between products of two primes.

infinitely-often （70）

Using we define prime equations,

（71）

From （71）we have

（72）

We redefine prime equations

（73）

From（73）we have

（74）

Goldston et. al proved only

infinitely-often [4-5]. （75）

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 , (http:// www. wbabin.net /math/xuan40.pdf).

[3] Chun-Xuan Jiang, Jiang’s funciton 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, 98 (2009) 741-774.

[5] D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between prime or almost primes, Trans. Am. Math. Soc.,361(2009)5285-5330.

美国匈牙利土耳其数学家正在研究和两个数。有无限多个使得和两个数每个都是两个素数相乘[4]。例如，，，,这就是当代国际数学最高水平, 受到当代数学界的关注。他们并没有证明这个问题, 它比哥德巴赫猜想难一万倍。蒋春暄看到[4] 以后, 国外就这么点水平吹到天上去, 决定写本文, 在国内外散发。蒋春暄2002年结果[1]。从定理一得出，，，有无限多个使得，，三个数每个数都是两个素数相乘。从定理二得出，，，，有无限多个使得每个数都是个素数相乘。定理六，，，，，有个数,有无限多个使得每个数都是个素数相乘，这样成果在过去没有数学家想象过，这是素数分布一个重要规律。将来一定会有广泛的应用。这是数学美！这是人类数学中最伟大成就, 在中国被评为最大伪科学, 中国不承认蒋春暄成就。It was therefore a great surprise for Erdos (and pobably for other number theorists as well) when C.Spiro proved in 1981 that d(x)=d(x + 5040) infinitely-often. It is theorem 11.There cannot modern prime theory without Jiang function.

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