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 白涛 （bxf77330@yahoo.com.cn） 2008.04.23 11:19:37
Brief Proof to Goldbach Conjecture and Twin
Prime Problem by Applying “Periodic Distribution Theorem of Prime Number” and “Product
Prime Number Distribution Density Theorem”
Bai Tao
Langfang Radio & TV University, Hebei province, China
（January
22, 2008）
Abstract:
the
thesis briefly proves Goldbach Conjecture and Twin Prime Problem by applying “Periodic
Distribution Theorem of Prime Number” and “Product Prime Number Density
Theorem” discovered by the writer.
Key words: Periodic Distribution Theorem
of Prime Number and Product Prime Number Density Distribution Theorem, periodic
law of odd3 odd number, Goldbach Conjecture, Twin Prime Problem
Goldbach
Conjecture and Twin Prime Problem can be briefly proved by applying the “Periodic Distribution
Theorem of Prime Number” and “Product Prime Number Distribution Density Theorem”
discovered by the writer.
[Note 1]
The “Periodic
Distribution Theorem of Prime Number” refers that, in every group of ten consecutive
positive integers (hereof refer as “10-order integer”) whose numeric position
is different in ones place but same in other numeric position, there is “periodic
odd3 prime number” formed by“periodic odd3 odd number” in existence for every three consecutive
positive integers. The number limit of the periodic difference is 30.
Here, “odd3
odd number” refers to “odd number positive integer of 10 equal order” ( hereof refer
as “10-order odd number”) excluding the ones divided exactly by 3 and 5. At the
same time, in the natural numbers, the n continuous “odd number of odd3” (n→∞) also forms a periodic “odd3
odd number cycle”, which can be named as “odd3 odd number periodic law”. Particularly,
the “odd3 odd number” can further form a periodic “odd3 odd number cycle” in which single prime number, dual prime number and twin prime number exist. The “dual
prime number” hereof refers to two prime numbers whose mantissas are in the
same 10 numeric position. While the “Twin Prime Number” refers to two prime
numbers, the difference between which is even 2. With the gradual expansion of
natural numbers, this“odd3 odd number cycle” shall be an infinite cycle.
According
to the “Theory of Numbers”, the positive integers whose mantissas are 5
and 0 can be divided exactly by 5. Thus, the integer whose “digital sum” of each
numeric position can be divided exactly by 3 (including 9) can be divided
exactly by 3 (including 9) [Note 2]; so it is certain to make the “odd number
of odd3” exclude integers which can be divided exactly by the 3 and 5. Odd3 odd
number has a fascinating property. Among the Odd3 odd number that consecutively
expand according to the natural order of natural numbers, would
automatically appear a ordered permutation and combination of odd numbers which
have a property of periodic cycle: among which, the first and the second
“ten-order odd number” (hereof refers as ten-order odd number one and two), each
have a group of “dual odd number” respectively，while the third “ten-order odd
number” (hereof refers as ten-order odd number three), is necessary to consist
of two groups of “dual odd number”. This kind of ordered permutation and
combination is invariable and unchangeable. Here, “Twin Odd Number” refers to the two
odd numbers whose difference is 2, while the “dual odd number” is two odd
numbers whose mantissas are in the same 10 numeric position. Moreover, from
n>20, the mantissas of “dual odd number” and “twin odd number” in each group
of odd3 odd number are the same as the ones of the “dual odd number” and “twin
odd number” in the correspond place of odd3 odd number in other groups. This is
because, the cycle of 3k, each multiple of three, is right the even number 30,
and odd3 odd number is exactly in each of its cycles, which exclude the
multiples of 3 for all values within a basic unit ten. Therefore, the rest odd
numbers are all “dual odd numbers” & “Twin Prime Numbers” corresponding in
pairs with the numerical value difference of 30 in odd3 odd numbers of two consecutive groups. This is the basic reason that odd3 odd number can form the periodic odd3 prime number.
The “periodic
odd3 odd number” reflects the law of an internal ordered permutation of natural
number.
Example 1: periodic
odd3 odd number (natural sequence n>20):
（23，29）；（31，37） ；
【（41，43），（47，49）】
（53，59）；（61，67） ；
【（71，73），（77，79）】
（83，89）； （91，97） ；
【 (101, 103) ， (107, 109)】
（113，119 ）； (121，127) ; 【(131，133 )，（137,139) 】
（143，149）； (151，157) ; 【(161，163) ，（167，169)】
（173,179） ； (181 ，187
); 【(191, 193) ， (197,199)】
（203，209）； (211，217) ; 【(221, 223) ， (227,229)】
（233,239） ；
(241,247) ; 【(251，253)
， （257,259)】
（263,269） ；
(271,277 ); 【(281,283) ， (287,289)】
（293,299） ；（301，307）; 【(311,313)
， (317,319)】
（323，329）；
(331，337) ; 【(341，343) ，（347，349)】
（353，359） ； (461，367); 【(371，373) ， (377，379)】
（383，389） ；（391，397）; 【(401，403) ，(407，409)】
-------------------------------------------------------
Thus we can
see that, the so called odd3 odd number is completely composed by single prime
number and multiple of the single prime numbers. The multiple of the single prime
number is also called “product prime number” (the product of two prime numbers).
They can be called “generalized prime number”. Among them, the reason why “ten-
order odd number three” has 4 generalized prime numbers
is that the
three “ten-order odd number” in each group of odd3 odd number includes five odd
numbers respectively. Since there must be one “ten-order odd number” whose multiple
of three is an even number and only one multiple of five, so the rest four are
generalized even numbers. Also the cycle of odd3 odd number starts from the natural
sequence
n>20, so it is “ten-order odd number three”.
From odd3 odd
number, we get “periodic odd3 prime number”; the numerical value range of the periodical
interval is also 30.
Sample 2: periodic
odd3 prime number (natural sequence n>20)
（23，29）；（31，37）；
【（41，43）；47】
（53）；
（61，67）；
【（71，73）；79】
（83，89）； （97）
；
【
(101,103)；(107,109)】
（113）
；
(127) ; 【131；（137,139)
】
（149）
；
(151，157)
; 【163，167】
（173,179）；
(181 ) ; 【(191,193);(197,199)】
(211) ; 【
223; (227,229)】
（233,239）；
(241 ); 【251；257】
（263,269）；
(271,277); 【(281,283);(289)】
（293）
；（301，307 ）;
【(311,313);317】
（329）： (331，337 )；
【347， 349】
（353，359）； (361，367);
【373； 379】
（383，389）；（397）
; 【401， 409】
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----------------------------------------------------------------
In the second
example, numbers in the square brackets are Twin Prime Number or dual prime
number, numbers in the parenthesis are dual prime number or single prime
number, it is apparently that “odd3 prime number” also reflect an internal law
of ordered arrangement of the natural number.
From the above
mentions we know actually that the “periodic distribution theorem of prime
number” is a theorem that concerning the reconfirm of the measurement unit of
the natural number. It takes 10 instead of 1 as the basic unit for measuring
the natural number, and divides all the natural number into consecutive number groups
of natural increase, which takes 10 as its basic unit, and then gets numberless
“10-order integer”. In fact, the “periodic distribution theorem of prime
number” adopts the “scale-of-thirty”，for only in this way, the relevant odd3 odd number or
odd3 prime number as well as its periodic circulatory distribution regularities
in the natural number can be found.
Prove: “periodic
distribution theorem of prime number”
Before the
proof, we first quote the “product prime number distribution density theorem”
discovered by the author of this text.
The “product
prime number distribution density theorem” is: in any finite intervals of the
natural numbers, several product prime number are distributed, and their
numbers shall be related to the distribution density of the product prime
number in this interval, which can be called the “average distribution density”
Z.
The “maximum distribution
density of odd3” Zn of product prime number, is the ratio between the “minimum interval” So of the adjacent two minimum product prime number and the “10
order odd3” interval Rn of odd3 in the same group, it is: Zn= Rn/ So.
The so-called
“minimum product prime number” means the product prime number obtained by
multiply any two minimum single prime number within 100. The difference value
between them is So. For example, the minimum interval of the product prime
number formed by the difference value between 31×13 and 37×11 is So=407—403=4. What shall be
pointed out is that So shall absolutely not 2. this is because that every product
prime number is composed by the sum of a single prime number which is much
smaller and the even number 2, the number of which is n. The number of these
even number can not only be 2n+1 (odd number), but also be 2n (even number). Although
the product of such two product prime number is composed by the sum of a single
prime number which is much smaller and even number 2, the number of which is n,
the number of these even number shall be 2n (even number). So the difference
value between the two product prime number shall not be the even number 2. Herefrom,
got the maximum distribution density of the product prime number Zn= Rn / So
=10/4=2.5, but can not got the integer 3. Then according to the average interval
of the product prime number in Rn is Sp=2+6/2=5, got the “odd3 average
distribution density” Zp=Rn/Sp=10/5=2, so, in the interval
of 2n, there are two product prime number at the very most.
The above is
the “product prime number distribution density theorem”.
Because in the
interval Rn of each odd3 odd number, there are 2 product prime number at the
very most. However, because in each group of Rn, there are always 4 generalized
prime numbers, therefore, in the odd3prime number cycle, there are the single
prime number periodic cycles, and the periodic interval is 30. This is the “periodic
distribution theorem of prime number”
Finished.
Proof to
Goldbach Conjecture and Twin Prime Number Problem
(1) The reasonableness
of Goldbach Conjecture
It can be
known from the “Distribution Theorem of Prime Cycle” that there shall be at least two
single prime numbers in the minimum numerical interval Rn of every group of
prime number odd3. However, only the fixed single prime number of 1, 3, 5, 7,
11, 13, 17, 23, and 29 are included in the numerical interval
30 for each group of the odd number odd3. Thus, all the even numbers within 30 in the numerical interval of each group of odd number odd3 can be obtained by the two prime numbers in each group of odd number odd3 plus or minus any of the above
mentioned fixed single number. Moreover, since the odd number odd3 has the
property of periodic cycle, all even numbers in natural numbers can be obtained
by analogy. So it is reasonable for Goldbach Conjecture.
Finished.
(2) The twin prime problem is only reasonable for including dual prime number.
It can be
known from the “Distribution Theorem of Prime Number Density” that the single prime number in
numerical interval Rn shall be at least two. At the same time, there shall be
two twin odd numbers in Rn. Since the minimum numerical interval So of product prime number is
4 and the numerical interval of twin odd number is 2, there must be dual prime
number when two single primes are left, and a pair of twin prime number when
three single prime numbers are left. Since the odd number odd3 has the property
of periodic cycles, there will be infinite twin prime numbers and dual prime
numbers. Hence, Twin prime number problem is only reasonable for including dual prime
number. That is, there is a periodic cycle for two dual prime numbers whose
difference is an even number of 2, 4, 6, and 8 in natural numbers.
Finished
【Note 1】大科普网 >
→ 数学 > 数学分支巡礼
> → 数论 “数学中的皇冠——数论”
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【Note 2】“ 公务员招考行测辅导：数字的整除特性”。中国教育在线 ——公务员频道。http://gongwuyuan.eol.cn
2007.10.16
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