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人类第一次打开素数大门（III）
蒋春暄
在素数大门有两个猜想：孪生素数猜想和哥德巴赫猜想。到今天仍没有解决，那么素数分布研究成果几乎等于零。陶哲轩和格林证明了存在任意长的素数等差数列。陶哲轩获得2006年国际数学家大会菲尔茨奖。加拿大著名数论专家Andrew Granville对陶哲轩和格林工作评价：There are certain cases that they are not able to attack as yet:
that there are infinitely many pairs of primes
The Simplest Proofs of Both Arbitrarily Long Prime Arithmetic Progressions
Chun-Xuan Jiang
P. O. Box 3924, Beijing 100854
P. R. China
Jiangchunxuan@sohu.com
Abstract
We define arbitrarily long prime arithmetic
progressions:
In prime numbers theory there are both well-known
conjectures that there exist arbitrarily long prime arithmetic progressions. In
this paper using Jiang functions
Theorem 1. We define prime arithmetic progressions:
We rewrite (1)
We have Jiang function [1]
where
From (4) we have
We prove that there exist infinitely many
primes
We have the best asymptotic formula [1]
where
Substituting (5) and (7) into (6) we have the best asymptotic formula
From (8) we are able to find the smallest
solution
Grosswald and Zagier obtain heuristically even asymptotic formulae [2]. Green and Tao obtain only lower bound [3].
Example 1.
Let
From (5) we have
We prove that there exist infinitely many
primes
Example 2.
Let
From (5) we have
We prove that there exist infinitely many
primes
Example 3.
Let
From (5) we have
We prove that there exist infinitely many
primes
Theorem 2. We define another prime arithmetic progressions[1, 4]:
where
We have Jiang function [1, 4]
where
If
If
Let
From (21) we have [1, 4]
We prove that there exist infinitely many
primes
We have the best asymptotic formula [1, 4]
Substituting (7) and (23) into (24) we have
From (25) we are able to find the smallest
solutions
Example 4.
Let
From (23) we have
We prove that there exist infinitely many
primes
Example 5.
Let
From (23) we have
We prove that there exist infinitely many
primes
Example 6.
Let
From (23) we have
We prove that there exist infinitely many
primes
From (34) we are able to find the smallest
solutions
Acknowledgement the Author would like to thank Zuo Mao-Xian for helpful conversations.
References
[1] Chun-Xuan, Jiang, Foundations of Santiili’s isonumber theory with applications to new cryptograms, Fermat’s theorem and Goldbach’s conjecture, Inter. Acad. Press, 2002, MR 2004c: 11001, www.i-b-r.org
[2] E. Grosswald, Arithmetic progressions that consist only of primes, J. Number Theory, 14, 9-31 (1982).
[3] B. J. Green and T. C. Tao, The primes contain arbitrarily long arithmetic progressions, to appear in Ann. Math.
[4] Chun-Xuan, Jiang, On the prime number theorem in additive prime number theory, Yuxinhe mathematics workshop, Fujian normal university, October 28 to 30, 1995.
只花了半天完成这个定理1证明 2006年10月5日
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蒋春暄 （jiangchunxuan@sohu.com） 上传2009.05.05
浏览243
(the twin prime conjecture); for any
large even integer 
there are pairs of primes 
(the
Goldbach conjecture), and that there infinitely many pairs of primes 
(Sophie
Germain twins). 他们不能证明孪生素数猜想和哥德巴赫猜想，他们只能在素数大门外瞎猜。2006年9月25日<科学时报>王元对陶哲轩工作评价：“我不敢想象天下会有这样伟大的成就。”我们看到陶哲轩和格林65页的论文，他们只得出一个下限，谁是变量也没有交待。2006年10月5日我们花了半天证明本文定理1。证明了存在任意长的素数等差数列，只用8个公式就证明了定理1。公式（5）证明了存在无限多素数
和
使得
都是素数。公式（8）得出计算
. Using Jiang
function 
we
prove that there exist infinitely many primes 
. Using Jiang function 
we prove
that there exist infinitely many primes 
are all primes and
find the best asymptotic formula (25).
. (1)
.
（2）
.
（3）
denotes the number of solutions
for the following congruence
（4）
as 
.
（5）
.
（6）
,
（7）
is called primorials, 
Euler
function.
.
（8）
for
large 
.
.
From (2) we have
. （9）
as 
are
primes. From (8) we have the best asymptotic formula
. （11）
.
From (2) we have
（12）
as 
are
all primes. From (8) we have the best asymptotic formula
.
（14）
.
From (2) we have
,
.
（15）
as 
are
all primes. From (8) we have the best asymptotic formula 
. （17）
（18）
is called a common difference, 
th prime.
,
（19）
（20）
.
, then 
; 
otherwise. From (20) we have
.
（21）
then 
, 
, there exist finite primes
are
all primes. If 
then
there
exist infinitely many primes 
.
（22）
, as 
.
.
（24）
(25)
for
large 
.
, 
, 
. From (22) we
have the twin primes theorem
（26）
as 
.
（28）
, 
. From (22) we
have
（29）
as 
.
（31）
, 
, 
. From (22) we
have
（32）
as 
are
all primes. From (25) we have the best asymptotic formula
. （34）
.