New Computing Model of $GN_{e}TM$ Turing Machine On Solving Even Goldbach Conjecture

The theory research about even Goldbach conjecture and the computer search has obtained significant achievement in many studies [1-9], but the research of its last certify is little progress. In artificial intelligence age and recent years, we have been known that Simpleminded AlphaGo of powerful intelligence computing wins by a high score to get the better of Legendary players [10], it not only to shock the whole world but also opened a new research idea to us. Is that the computer can solve the problem of the even Goldbach’s conjecture? And any given an even number, the computer within finite step whether or not calculable, this is people very concerned one question. If the problem of even Goldbach conjecture is the computer recursion solvable, then it’s solution model how to correct description? And how to confirm even Goldbach conjecture whether to exist uniqueness judge result? It still can solve the problem of infinite computation by computer. The question of these key points is discussed in this paper.

Suppose the solution process of the computer for even Goldbach conjecture can be abstracted as new computing model of dream Turing Machine of a simplified deformation, and abbreviated to $G{{N}_{e}}TM$—-Goldbach Number Turing Machine,to use it distinguish the results of true (T) or false (F) about the prime matching existence. As shown in Fig 1.

The machine model is consist of some parts of the next contents:

1. A finite controller, i.e., a set of computing control rule $R{{N}_{e}}$ of the prime matching algorithm that to solve even Goldbach conjecture. It is according current read-write head to point at the tape lattice even number ${{N}_{ei}}\;(i=1,2,\cdots)$ and present machine run to be in the state ${{q}_{i}}$ of computation result, in turn, determined read-write head to execute next run operation, and changing the result (T/F) of the status register, even which makes the machine change over to the next new status.

2. An input tape of has infinite tape lattice. The left endpoint is initiation point, right endpoint point to infinite. In each tape lattice just to denote the number of the even set $\{4,6,8,\cdots\}$ that from the number 4 starting. But there is not the number 0 and 2, as well as a special blank mark$\Box$. From the left to the right, in turn, serial number is $1,2,3,\cdots$ in the tape.

3. A read-write head. It can along the left and the right to move in the tape, and to execute three action: first is to change determine the status of computing result in finite controller; second is to execute normal operation: the function of in order the right, it can read out corresponding even number in present tape lattice. And to execute abnormal operation: the function of in order to the left, and can repeat the operation, several time in succession readout corresponding even number in present tape lattice (use to check). The third is to execute the output result of controller status, and to print out true (T) or false (F) symbol in the corresponding tape lattice.

4. An output status register. It is used to save the computation result status of the present position in the Turing machine, i.e., ${{q}_{i}}\in\{T/F\}$. The numbers of all possible states in the model $G{{N}_{e}}TM$ is limited, it mainly indicates the correct status number of the prime matching and incorrect status number of the prime matching. The latter state is appointed to special halting status.

Here special provision that the halting status of $G{{N}_{e}}TM$ machine model refer to if the prime matching for even Goldbach conjecture is not established, then the computing result makes machine halting (but the process may repeat read the number and check the result). Otherwise, if the machine not halting, then indicate the result of the prime matching is established, and the machine continues to run to infinity.

The model $G{{N}_{e}}TM$ is detailed description as follow:

Definition 1.1. The computer solvable process of even Goldbach conjecture which assumes is called new computing model of dream Turing Machine about a simplified deformation, and abbreviated to $G{{N}_{e}}TM$. TM has a 7-Tuple:

$$\{QG{{N}_{e}},\Sigma{{N}_{e}},R{{N}_{e}},{{q}_{0}},\delta,\text{T},\text{F}\}$$

$QG{{N}_{e}}$: The infinite set of the non-empty status that about the result both of right and wrong of the prime matching computing $({{p}_{i}}+{{p}_{j}}={{N}_{e}})$ for even Goldbach conjecture. $\forall{{q}_{i}}\in QG{{N}_{e}},$ ${{q}_{i}}\;(i\geq 1)$ is a status of TM, It is except for original state, or as right status or as wrong status, ${{q}_{i}}\subseteq\{T/F\}$.

${{q}_{0}}$: an original status, ${{q}_{0}}\in QG{{N}_{e}}$. TM is run operation from ${{q}_{0}}$ state start, at the same time, the read-write head point to the leftmost even number in the tape.

$\Sigma{{N}_{e}}$: It is the even number set of finite input: $\{4,6,8,\cdots,{{N}_{e}}\}\;({{N}_{e}}=2n,n\geq 2)$, ${{N}_{e}}\in\Sigma{{N}_{e}}$. Only if the even number of given at $\Sigma{{N}_{e}}$ and $\Sigma{{N}_{e}}^{\infty}$, it can when TM starting to appear at the input tape. Among other excluding the number 0, 2, and a special blank symbols$\Box$; $\Sigma{{N}_{e}}^{\infty}$ is specific reference to input even number set of dream infinite $\{4,6,8,\cdots,{{N}_{e}},\cdots\}$.

$R{{N}_{e}}$: In the controller of $G{{N}_{e}}TM$, it expresses the rule set of the computing prime matching is the correct or not, and it is the finite matching computation of corresponding each ${{N}_{e}}$ in the independence closed.

$\delta$: The element $\delta:QG{{N}_{e}}\times R{{N}_{e}}\to QG{{N}_{e}}\times R{{N}_{e}}\times\{R,L\}$ is the transition function. Here R and L express read-write head are moved to the right or left.

T(true): It is the matching status that the result appears correct by controller computing, $T\in QG{{N}_{e}}$, ${{q}_{i}}\subseteq T\;(i\geq 1)$. i.e., the machine continues to run status.

F(false): It is the matching states that the result appears incorrect by controller computing, $F\in QG{{N}_{e}}$. If ${{q}_{j}}\notin T\;(j\geq 1)$, If only ${{q}_{j}}\subseteq F\;(j=1)$, then $G{{N}_{e}}TM$ is the termination status, i.e., the machine is halting status.

The run process of the operation status on $G{{N}_{e}}TM$ Turing Machine as shown in Fig 2.

According to the principle of $G{{N}_{e}}TM$ computing model, we further discussion above model whether exist thus a program (it is used to check the algorithm of the prime matching), it can computing given even $N{}_{e}$ in the input tap, and to can prove that the even Goldbach conjecture is all computer recursion that can be solved. Secondly, it as also can further to verify $\forall{{q}_{i}}\subseteq T\;(i\geq 1)$, $N{{T}_{i}}\;(i=1,2,\cdots)\in T$,the result is all true. And the machine can judge that $\forall{{N}_{e}}$ in the input tape which to satisfy the even Goldbach conjecture is all established. The machine not halting run to infinity. On the contrary, if to verify some status ${{q}_{j}}\subseteq F\;(j>1)$, and by repeat check $N{{F}_{j}}\equiv{{{N}_{e}}}/{2}\;\subseteq F$, the result is all false. Then the machine stops running and shows that it stops, which indicates that even Goldbach conjecture has not been established.

Aimed the questions raised above, we will depth one by one discusses it in next.

Definition 2.1. Let ${N}_{e}$ be any non-zero even number and ${N}_{e}=2n,n\geq 1$. Assume $x,y\in{{N}_{e}}$ is two positive integers of ${N}_{e}$. If the difference between ${N}_{e}$ and $y$ can be divided by $x$ in the integer field of ${N}_{e}$, then it call ${N}_{e}$ and $y$ are the congruence of module $x$, it is written as:

$$N_{e}\equiv y(mod\,x),\;(x,y\in N_{e})$$

(1)

In addition, if the difference between ${N}_{e}$ and $x$ can be divided by $y$, then it call ${N}_{e}$ and $x$ are the congruence of module $y$, and it is written as:

$$\centering N_{e}\equiv x(mod\,y),\;(x,y\in N_{e})\@add@centering$$

(2)

The formula (1) and (2) are collectively called the pairing formula of the even congruence relations.

Because the two formulas are similar, for the convenience of analysis, the following is discussed only in the case of (1).

Definition 2.2. It is called the operating of the congruence independent closure for ${N}_{e}$, it is referred to satisfy the situation ${{N}_{e}}\equiv{{y}}(\bmod\,x)$ $(x,y\in{{N}_{e}})$, the operating of all intermediate values and results are the range within ${N}_{e}$. i.e., $[x|({{N}_{e}}-y)]\equiv 1\to{{N}_{e}}|(x+y)\equiv 1$, and to satisfy the smallest positive residual relationship of the congruence by a given module $x$ (or given module $y$).

Obviously, the congruence independent closure in ${N}_{e}$, the definition 2.1 describes the expression relationship of add sum of two even numbers. Another, according to the theory of the congruence, we can easily get the following lemma.

Lemma 2.1. (the congruence expression theorem of implicit even sum ) Assume that ${N}_{e}$ is non-zero even number, ${{N}_{e}}=2n,n\geq 1$, and $x,y=2n-1,n\geq 1\;(x,y\in{{N}_{e}})$, and $x,y$ both are positive odd integer in ${N}_{e}$. In the independent closure of ${N}_{e}$, any given a positive even number ${{N}_{e}}$, it must exists two positive odd integers $x$ and $y$, and to satisfy the as following relation:

$${N}_{e}\equiv y(mod\,x)\;(x,y\in N_{e})$$

Proof. According to the congruence relation, as long as the independent closure within ${{N}_{e}}$ is given, two positive odd integers $x$ and $y$ will be found to form a congruence relation:

$$x\equiv-y(mod\,N_{e})\quad(x,y\in N_{e})$$

$$(or:-x\equiv y(mod\,N_{e})\quad(x,y\in N_{e})$$

Are established and satisfy $({{N}_{e}}|(|x+y|))\equiv 1$.

Now think about it the other way around, by definition 2.1, if only for the element $x$, swap ${N}_{e}$ with $x$, and y is fixed. Because $x$ and $y$ are both positive odd numbers in the element $x\not=y$ or $x=y$. According to the congruence relation, the following congruence expressions with even numbers can be summed, i.e.,

$$N_{e}\equiv y(mod\,x)\quad(x,y\in N_{e})$$

It is established, and satisfied the $[x|(({{N}_{e}}-y)=x)]\equiv 1$. For have $y={{N}_{e}}-x$, above result can be also expressed as:

$$N_{e}\equiv N_{e}-x(mod\,x)$$

And to satisfy the formula as: $[x|({{N}_{e}}-({{N}_{e}}-x))]\equiv 1$.

In fact, the lemma of this implicit expression even sum, which actually describes the establishment result of ${N}_{e}\equiv y+x$. The lemma shows that an even Goldbach Guess is the relationship between adding the sum of two odd numbers. Lemma 2.1 can also describe equivalently the existence relation model of even add sum within ${N}_{e}$ independent closure.

Definition 2.3. “ The odd complete congruence expression group of determinate module $x$ ” means that within the congruence independent closure of ${N}_{e}$, even number ${N}_{e}$ is the relationship formulas of the congruence sum of consisted of all enable it established full permutation for two odd positive integers element $x$ and $y$, i.e.,

$$\left\{\begin{array}[]{l}{{N}_{{e}}\equiv{y}_{{N}_{e}-1}\bmod{x}_{1}}\\ {N}_{{e}}\equiv{y}_{{N}_{e}-3}\bmod{{x}_{3}}\\ {\vdots}\\ {{N}_{{e}}\equiv{y}_{3}\bmod{x}_{{N}_{e}-3}}\\ {{N}_{{e}}\equiv{y}_{1}\bmod{x}_{{N}_{e}-1}}\end{array}\right.$$

And it’s written as:

$${{N}_{e}}{{(o)}^{{\;}}}{{\underline{{\bar{\equiv}}}}}{{Y}_{o}}^{{\;}}(\bmod\,{{X}_{o}})\quad(shortnote:Mod\,\overset{\equiv}{\mathop{X}}\,(o))\\$$

Lemma 2.2. Independent closure in ${{N}_{e}}$, “The odd complete congruence expression group of determinate module $x$ ” contains full permutation congruence expression of the odd integrity of ${{N}_{e}}$/2 pairs, and it satisfies the basic characters: uniqueness, closure, symmetry, constructive and expansible.

Proof. Selecting ${{k}}(1\leq k\leq{{N}_{e}}-1)$ as an enumerate factor within the congruence independent closure of ${{N}_{e}}$, after to act on $x$ and $y$ by $k$ separately, always existent respective formula ${{N}_{e}}\;{{\equiv}_{{\;}}}{{y}^{{}^{\prime}}}(\bmod\,{{x}^{{}^{\prime}}})$, and makes it is established. And the structure of full permutation congruence expression for the minimum positive remaining of all odd integer, which as showed in the definition 2.3. i.e.,

$${{N}_{e}}{{\;\overline{\underline{\equiv}}}^{{\;}}}{{Y}_{{{k}^{{}^{\prime}}}}}(\bmod\,{{X}_{k}})={{[{{N}_{e}}\equiv{{y}_{{{N}_{e}}-k}}{{\bmod\,}_{k}}]}_{(1\leq k\leq{{N}_{e}}-1)}}$$

Here ${{k}^{{}^{\prime}}}$ and $k$ are mutual duality relation. Obviously, through the determinate the congruence expressions shape of the integrity full permutation within the independent closure of ${{N}_{e}}$, when $x,y=2n-1,(n\geq 1)$, for $1\leq k\leq{{N}_{e}}-1$, as long as the enumerate of $k$ is not lack the number of items. According to the definition 2.1 and lemma 2.1, in the complete congruence expression group for all determinate module $x$, which must exist full permutation form of integrity congruence expressions of ${{N}_{e}}/2$ pairs. And it satisfies as following the trait.

Uniqueness. Given any even number ${{N}_{e}}$, and enumerating each number at $1\leq k\leq{{N}_{e}}-1$, the module $x$ from $1\to{{N}_{e}}-1$ enumerating, the element $y$ is corresponding $x$ by ${{N}_{e}}-1\to 1$ with it paired, and it satisfies the relation permutation of ${{[{{N}_{e}}\equiv{{y}_{{{N}_{e}}-k}}\bmod{{x}_{k}}]}_{(1\leq k\leq{{N}_{e}}-1)}}$. As long as ${{N}_{e}}$ is given, and the form that full permutation not lack the number of items of the odd integrity congruence expression group is also assured, then the uniqueness is permanently true.

Closure. Examining every congruence expression ${{N}_{e}}^{{\;}}{{\underline{{\bar{\equiv}}}}_{{\;}}}{{Y}_{{{k}^{{}^{\prime}}}}}(\bmod\,{{X}_{k}})$, ${[{N}_{e}\equiv{y}_{{{N}_{e}}-k}}$ mod ${{x}_{k}}]$, $(1\leq k\leq{{N}_{e}}-1)$ are all established, and it satisfy to $[{{x}_{k}}|({{N}_{e}}-{{y}_{{{N}_{e}}-k}})]\equiv 1$. According to the definition 2.2. All the congruence expressions are operating in the independent closure in ${{N}_{e}}$. Then, by lemma 2.1 can be known that its closure is really.

Symmetry. From the shape of the full permutation and the character of the uniqueness of the integrity congruence expression, it’s easy found the existence of the symmetry relationship in $Mod\,\overset{\equiv}{\mathop{X}}\,(o)$. Namely, ${{N}_{e}}\;\underline{\overline{\equiv}}\;{{Y}_{{{k}^{{}^{\prime}}}}}(\bmod\,{{X}_{k}})=$ ${{[{{N}_{e}}\equiv{{y}_{k}}\bmod\,{{x}_{{{N}_{e}}-k}}]}_{(1\leq k\leq{{N}_{e}}-1)}}$ is true too. If only $k$ just selecting to a half of the range can satisfy the symmetrical relationship.

Constructability. When ${{N}_{e}}$ is given, then ${{N}_{e}}^{{\;}}{{\underline{{\bar{\equiv}}}}_{{\;}}}{{Y}_{{{k}^{{}^{\prime}}}}}(\bmod\,{{X}_{k}})$ is constructed by the effect of the enumerate factor $k$. As long as enumerate $k$ makes all congruence expression group is established in the range of $1\leq k\leq{{N}_{e}}-1$, theoretically, in $\forall{{N}_{e}}$, we enumerating every ${{N}_{e}}$, it satisfies the construction of the full permutation of the odd integrity congruence expression group. It obviously establishes.

Expansible. If only the element $x,y\to\infty$,${{N}_{e}}\to\infty$, then $Mod\,{}\overset{\equiv}{\mathop{X}}\,(o)\to\infty$. The result must be existing expansible.

Deduction 2.1. Assume that $\phi(Mod\,\overset{\equiv}{\mathop{X}}\,(o))$ is expressed as the numbers of full arrangement pairs (allowing equivalent repeatable) that the congruence relation consist of all odd number ($x,y$) to exist all determined modulo x in $Mod\,\overset{\equiv}{\mathop{X}}\,(o)$, then there is $\phi(Mod\,\overset{\equiv}{\mathop{X}}\,(o))={{N}_{e}}/2$.

Proof. According to the definition 2.3 and the lemma 2.2, the deduction established is can be proved.

Definition 2.4. “The congruence expression group of the prime of determinate module $x$” means that in the independent closure of ${{N}_{e}}$, the element $x,y$ are all consist of the prime $(p{{{}^{\prime}}_{1}},p{{{}^{\prime}}_{2}},p{{{}^{\prime}}_{3}},\ldots,p{{{}^{\prime}}_{s-2}},p{{{}^{\prime}}_{s-1}},p{{{}^{\prime}}_{s}})$ $(1\leq s<r)$, it is the set of the congruence add sum expression of a set of the permutation relation in determined module $x$. It is written as follow:

$${{N}_{e}}{{(p)}^{{\;}}}{{\underline{{\bar{\equiv}}}}_{{\;}}}Y{{{}_{p}}}(\bmod\,X_{p})\quad(shortnote:Mod\overset{\equiv}{\mathop{X}}\,(p))$$

In here, $Mod\,\overset{\equiv}{\mathop{X}}\,(p)$ $\in Mod\,\overset{\equiv}{\mathop{X}}\,(o)$. Because of $Mod\,\overset{\equiv}{\mathop{X}}\,(p)$ in this definition, among the element $x,y$ are all consist of the primes ${{p}_{i}}$ $(1\leq i\leq r)$, but it must satisfy the result establish of the formula: ${{X}_{p}}+{{Y}_{p}}={{N}_{e}}(p)$, it’s not all prime in ${{N}_{e}}$ can be constituted the matching and to can satisfy this relation of even add sum.

Definition 2.5. “The prime congruence formulas of regular determinate module $x$” are expressed as the set of the congruence formulas of the prime universal arrangement in modulo $x$, which element can construct a group of non-repeating relation result in $Mod\,\overset{\equiv}{\mathop{X}}\,(p)$. It is written as the formula as following:

$$N_{e}^{+}{{(p)}^{{\;}}}{{\underline{{\bar{\equiv}}}}_{{\;}}}Y{{{}_{p}^{+}}}(\bmod\,X_{p}^{+}),(shortnote:Mod\,\overset{\equiv}{\mathop{{{X}^{+}}}}\,(p))$$

Here, $Mod\,\overset{\equiv}{\mathop{{{X}^{+}}}}\,(p)$ is also a kind of set that corresponding positive congruence residue system in the independent closed module $x$, among the element $x,y$ are all the prime and one to one non-repeating composing the congruence relation result. Therefore, $Mod\,\overset{\equiv}{\mathop{{{X}^{+}}}}\,(p)$ without the symmetry, but it still satisfies uniqueness, closed and constructible, as well as Expansible. So the relation as follow:

$$Mod\,\overset{\equiv}{\mathop{{{X}^{+}}}}\,(p)\in Mod\,\overset{\equiv}{\mathop{X}}\,(p)\in Mod\,\overset{\equiv}{\mathop{X}}\,(o)$$

$$Mod\,\overset{\equiv}{\mathop{{{X}^{+}}}}\,(p)\in Mod\,\overset{\equiv}{\mathop{{{X}^{+}}}}\,(o)\in Mod\,\overset{\equiv}{\mathop{X}}\,(o)$$

Lemma 2.3 (Chinese remainder theorem). Assume that ${{m}_{1}},{{m}_{2}},\cdots,{{m}_{r}}$ are r positive prime of the coprime numbers, ${{b}_{1}},{{b}_{2}},\cdots,{{b}_{r}}$ are any r positive integers, then have r congruence equations:

$$S\equiv b_{j}(mod\,m_{i})\quad(1\leq i,j\leq r)$$

Corresponding the module $m={{m}_{1}}{{m}_{2}}\cdots{{m}_{r}}$, it has the result of unique one solution. The solution is:

$$S\equiv M_{1}^{{}^{\prime}}{{M}_{1}}{{b}_{1}}+M_{2}^{{}^{\prime}}{{M}_{2}}{{b}_{2}}+...+M_{r}^{{}^{\prime}}{{M}_{r}}{{b}_{r}}(\bmod\,m)\equiv\sum\limits_{i,j=1}^{r}{M_{i}^{{}^{\prime}}{{M}_{i}}{{b}_{j}}}(\bmod\,m)$$

Where $m=m{}_{1}m{}_{2}...m{}_{r}$,${{M}_{i}}={}^{m}/{}_{{{m}_{i}}}$,$M_{i}^{{}^{\prime}}{{M}_{i}}\equiv 1\left(\bmod\,{{m}_{i}}\right)$,$M_{i}^{{}^{\prime}}=M_{i}^{-1}\left(\bmod\,{{m}_{i}}\right)$, $(1\leq i\leq r)$.(Proof omitted)

Lemma 2.3 describes the summation problem of r unknown congruent equations, where ${{b}_{j}}$ and ${{m}_{i}}$ is known, and $S$ is unknown. If now let $S$ and ${{m}_{i}}$ is known, ${{b}_{j}}$ is unknown, then the equation becomes to solve that the problem of ${{b}_{j}}\;(1\leq j\leq r)$ whether or not can satisfy the solution of the equation. So a new definition is given as follow:

Definition 2.6. In the matching formula of the congruence relations of even add sum, if ${{N}_{e}}=2n,n>2$ is any non-zero even number, let ${{m}_{i}}={{p}_{i}}$ $(1\leq i\leq r)$, $2<{{p}_{i}}\leq{{N}_{e}}-1$, it is any a prime within all odd prime numbers of less than ${{N}_{e}}$, and ${{b}_{j}}\;(1\leq j\leq r)$ is the positive integers of even congruence matching relations in closed ${{N}_{e}}$. If in closed even ${{N}_{e}}$, selecting every ${{m}_{i}}$ is all have unique ${{b}_{j}}$ corresponds to it, and have all formulas of congruence relation of even matching as follow:

$$\bmod\overset{\equiv}{\mathop{M}}\,({{N}_{e}})=\left\{\begin{matrix}{{N}_{e}}\equiv{{b}_{1}}(\bmod\,{{p}_{1}})\\ {{N}_{e}}\equiv{{b}_{2}}(\bmod\,{{p}_{2}})\\ ...\\ {{N}_{e}}\equiv{{b}_{r}}(\bmod\,{{p}_{r}})\\ \end{matrix}\right.\\$$

(Or it’s written as: $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ = [${{N}_{e}}$ $\equiv{{b}_{j}}(\bmod\,{{p}_{i}})(1\leq i,j\leq r)],({{b}_{j}},{{p}_{i}}\in{{N}_{e}})$) Then it is called extended remainder theorem equations, and short note: $\bmod\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$.

The difference between extended remainder theorem equations $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ with lemma 2.3 are that, ${{N}_{e}}$ and ${{m}_{i}}={{p}_{i}}$ are known, ${{b}_{j}}$ is unknown, where ${{b}_{j}}={{N}_{e}}-{{p}_{i}}\;(1\leq i,j\leq r)$, and each formula can be obtained result through specific solution and computing.

In fact, the definition 2.6 is another form of the definition 2.3, that is the module is the odd number becomes the module is the prime, both relation are $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ $\in$ $Mod\,\overset{\equiv}{\mathop{X}}\,(o)$. In the definition 2.6, the element $x\to p$,$y\to b$, x is to consist of all prime ${{p}_{i}}\;(1\leq i\leq r)$ within ${{N}_{e}}$, another y consist of non-deterministic corresponding quasi-prime $p_{i}\;(1\leq i\leq r)$ of equal number, $p_{i}^{{}^{\prime}}$ is the prime or the odd. In the independent closed ${{N}_{e}}$, practical two prime add result can make the relation of the even sum is established combination pair number, which at most for only $s$ $(1\leq s<r)$ pairs. Whether or not each event in ${{N}_{e}}$ at least all has one pair establishes, now to need we further continue verifying it.

And due to the relation of the definition 2.6, $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})\in Mod\,\overset{\equiv}{\mathop{X}}\,(o)$, therefore have:

$$Mod\overset{\equiv}{\mathop{X}}\,(p)\in\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})\in Mod\,\overset{\equiv}{\mathop{X}}\,(o)$$

First convention: It can be expressed as even number ${{N}_{e}}$ of two prime add sum called Goldbach number, which is written as $G({{N}_{e}})$. For the convenience of discussion, some concepts are proposed as follows:

Definition 3.1. Assume ${{F}_{1}}=\{{{O}_{s}},{{O}_{is}},{{S}_{o}}\}$ be a class of the patterns set for two odd number sequence corresponding each element arrangement composed of computing add sun in triples element $\{{{O}_{s}},{{O}_{is}},{{S}_{o}}\}$, where ${{O}_{s}}=\{1,3,5,\cdots,2n-5,2n-3,2n-1\}$ is a permutation set of odd number sequence (from small to large array) within any given even number ${{N}_{e}}=2n\;(n>1)$, ${{O}_{is}}=\{2n-1,2n-3,2n-5,\cdots,$$5,3,1\}$ is a permutation set of reverse odd number sequence (from large to small array) within any given even number ${{N}_{e}}=2n\;(n>1)$, and ${{S}_{o}}=\{(1+(2n-1))=2n,(3+(2n-3))=2n,\cdots,$ $((2n-3)+3)=2n,$ $((2n-1)+1)=2n\}$ is the set of even sum that corresponding to each group two element add sum operation.

Definition 3.2. Assume ${{F}_{2}}=\{{{P}_{s}},P_{is}^{{}^{\prime}},{{G}^{{}^{\prime}}}(N{}_{e})\}$ be a class of the set for the prime and quasi-prime (where have prime or odd number) two sequence composed of corresponding element permutation and each group two element add sun in triples element $\{{{P}_{s}},P_{is}^{{}^{\prime}},{{G}^{{}^{\prime}}}({{N}_{e}})\}$, where ${{P}_{s}}=\{{{p}_{1}},{{p}_{2}},{{p}_{3}},...,{{p}_{r-2}},$${{p}_{r-1}},{{p}_{r}}\}$ is a permutation set of all prime sequence (from Small number to large number array) within any given even number ${{N}_{e}}=2n\;(n>1)$, $P_{is}^{{}^{\prime}}=\{p_{r}^{{}^{\prime}},p_{r-1}^{{}^{\prime}},p_{r-2}^{{}^{\prime}},...,p_{3}^{{}^{\prime}},p_{2}^{{}^{\prime}},p_{1}^{{}^{\prime}}\}$ is a permutation set of reverse quasi-prime sequence (from large number to small number array) within any given even number ${{N}_{e}}=2n\;(n>1)$, and ${{G}^{{}^{\prime}}}({{N}_{e}})=\{({{p}_{1}}+p_{r}^{{}^{\prime}})={{N}_{e}},\;({{p}_{2}}+p_{r-1}^{{}^{\prime}})={{N}_{e}},\cdots,$ $({{p}_{r-1}}+p_{2}^{{}^{\prime}})={{N}_{e}},\;({{p}_{r}}+p_{1}^{{}^{\prime}})={{N}_{e}}\}$ is an array set of corresponding to each group of prime and quasi-prime both two element add sum, which is also called quasi- Goldbach number.

Definition 3.3. Let $\varphi({{S}_{o}})$ be the numbers of the combination pair of two element add sum for any given even number in ${{F}_{1}}$, $\phi({{G}^{{}^{\prime}}}({{N}_{e}}))$ be the numbers of the combination pair of two element sum relation for any given even number in ${{F}_{2}}$, and $\varphi(G({{N}_{e}}))$ be the numbers that satisfy Goldbach number combination pair of two element add sum in ${{F}_{2}}$.

Further analysis shows that ${{F}_{1}}\neq{{F}_{2}},\;{{F}_{2}}\in{{F}_{1}},\;{{N}_{e}}=2n,\;G({{N}_{e}})\;\in{{G}^{{}^{\prime}}}({{N}_{e}}),\;\varphi({{S}_{o}})\equiv n,\;\varphi(G({{N}_{e}}))\in\varphi({{G}^{{}^{\prime}}}({{N}_{e}})),\;\varphi({{G}^{{}^{\prime}}}({{N}_{e}}))=r,\;\varphi(G({{N}_{e}}))$ are unknown. If $\varphi(G({{N}_{e}}))=0$ (only if can find out an example), then the even Goldbach conjecture is not established. Conversely, if $\varphi(G({{N}_{e}}))\neq 0$, and can ensure $\forall{{N}_{e}},$ $\varphi(G({{N}_{e}}))=1$, then the even Goldbach conjecture is established. This paper will further discuss $\varphi(G({{N}_{e}}))=?$

According to the definition 2.6, a new description model $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ of equivalent existence for even Goldbach conjecture has been established. By calculating the probability of ${{q}_{j}}$ and matching to determine the probabilities, we verified the general conjecture criterion of even number conjecture.

Theorem 3.1. Even Goldbach Guess can be implicitly described by $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ model.

Proof. By the definition 2.1 and the lemma 2.1, as well as the definitions 2.2 and 3.2, we have known that $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ is an extended remainder theorem equation. If ${{N}_{e}}$ is given, in the independent congruence closure of ${{N}_{e}}$, the module ${{m}_{i}}$ $(1\leq i\leq r)$ is given, too. According the relationship of the even sum, it’s easy to obtain the corresponding ${{b}_{j}}$, for ${{N}_{e}}-{{m}_{i}}={{b}_{j}}$. Now let’s assume ${{m}_{i}}={{p}_{i}}$ $(1\leq i\leq r)$ be any element of the prime set within ${{N}_{e}}$, and let ${{m}_{i}}={{p}_{i}}{{,}}{{m}_{j}}={{p}_{j}}$, $({{p}_{i}},{{p}_{j}})=1$ be relatively prime. Furthermore, let ${{b}_{j}}={{q}_{j}}\;(1\leq j\leq r)$ be any odd of among odd integers set that makes two element add sum relation can established of corresponding to module ${{m}_{i}}={{p}_{i}}$ in ${{N}_{e}}$, ${{q}_{j}}$ be odd (${{q}_{j}}=2n-1$, $n\geq 1$) either prime (${{q}_{j}}={{p}_{i}}$, ${{q}_{j}}\neq{{p}_{i}}$). The result is:

$$\left\{{\begin{array}[]{*{20}{c}}{{N_{e}}\equiv{q_{1}}(\bmod\,{p_{1}})}\\ {{N_{e}}\equiv{q_{2}}(\bmod\,{p_{2}})}\\ {...}\\ {{N_{e}}\equiv{q_{r}}(\bmod\,{p_{r}})}\end{array}}\right.$$

And it can also be written as: ${{N}_{e}}\equiv{{q}_{j}}(\bmod\,{{p}_{i}})\;(1\leq i,j\leq r),({{q}_{j}},{{p}_{i}}\in{{N}_{e}})$

We use mathematical induction to prove that the result is established. When ${{N}_{e}}$= 6, then $6\equiv{{3}}(\bmod\,3)$ is established. When ${{N}_{e}}=m=$ $2n+2\geq 6\;(n\geq 2)$, then $m\equiv{{q}_{mj}}(\bmod\,{{p}_{mi}})$ $(1\leq i,j\leq r)$, $({{q}_{mj}},{{p}_{m}}_{i}\in m\;=\;{{N}_{e}})$ are established. Furthermore, when ${{N}_{e}}=m+2$, $(m=2n+2\geq 6,n\geq 2)$, then $m+2\equiv{{q}_{(m+2)j}}(\bmod\,p{}_{(m+2)i})$ $(1\leq i,j\leq r)$, $({{q}_{(m+2)j}},\;{{p}_{{{(m+2)}}i}}$ $\in m+2={{N}_{e}})$ is established.

Obviously, this structure model gives a new implicit description model of even Goldbach Guess, and each ${{N}_{e}}$ is all may be used the structure to describe it. If only ${{N}_{e}}$ is given, it is can be any enumeration. Once ${{p}_{i}}\;(1\leq i\leq r)$ is determined, ${{q}_{j}}\;(1\leq j\leq r)$ by solution result can also get determined corresponding element. Because ${{q}_{j}}={{N}_{e}}-{{p}_{i}}\;(1\leq i,j\leq r)$ can be determined to be an odd or a prime number, so we use the model $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ to describe the Even Goldbach Guess is also unique.

Now for example are described below, suppose ${{N}_{e}}=34$, the prime set $P={\{3,5,7,11,}$ ${13,17,19,23,29,31\}}$ within ${{N}_{e}}$, then have odd integers set $Q={\{31,29,27,23,}$ ${21,17,15,11,5,3\}}$ of corresponding in ${{N}_{e}}$. The form of $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ is given as following:

$$\bmod\mathop{M}\limits^{\equiv}(34)=\left\{{\begin{array}[]{*{20}{c}}{34\equiv 31\bmod{3}}\\ {34\equiv 29\bmod{5}}\\ {34\equiv 27\bmod{7}}\\ {34\equiv 23\bmod 11}\\ {34\equiv 21\bmod 13}\\ {34\equiv 17\bmod 17}\\ {34\equiv 15\bmod 19}\\ {34\equiv 11\bmod 23}\\ {34\equiv{5}\bmod 29}\\ {34\equiv{3}\bmod 31}\end{array}}\right.$$

Now we need to solve a question , if $\forall{{N}_{e}}$, each ${{N}_{e}}$ corresponding the element ${{q}_{j}}$ within Q ($({{q}_{j}}\in Q(1\leq j\leq k,Q\in{{N}_{e}}))$ sure exist a prime, and satisfy the relation of ${{q}_{j}}={{N}_{e}}-{{p}_{i}}\;(1\leq i,j\leq r)$, then the result definitely at least have a formula satisfy the requirement of even Goldbach conjecture in $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$. Thus it’s easy to verify the result of even Goldbach conjecture.

According to the definition 3.2, the theorem 3.1, and the analysis of the construction trait for the equation group $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ of Expanded Chinese Remainder Theorem(ECRT) in the definition 2.6, we have been found that, for even Goldbach conjecture can be expressed using the model $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$. When given any even number ${{N}_{e}}=2n,n>2$, within the independence closed range of every ${{N}_{e}}$, the odd prime set $P=\{{{p}_{i}}\}\;(1\leq i\leq r)$ is defined, too. For all odd prime not greater than ${{N}_{e}}$ as follow:

$$3={{p}_{1}}<{{p}_{2}}<{{p}_{3}}<\cdots<{{p}_{r}}<{{N}_{e}}$$

With it correspondent odd integer set $Q={{q}_{j}}\;(1\leq j\leq r=k)$ is also can be determined uniquely, because of ${{q}_{j}}={{N}_{e}}-{{p}_{i}}\;(1\leq i,j\leq r)$. Now the key question is whether we can prove that at least one of the elements ${{q}_{j}}$ in $Q$ is a prime number? and satisfy the relation result of ${{q}_{j}}={{N}_{e}}-{{p}_{i}}\;(1\leq i,j\leq r)$. That is to say, If its existence can prove to be a definite truth, then the even conjecture question can be solved. On the point, first, we further analysis the model described in the definition 3.2, 3.3 and the theorem 3.1, and for it use simplifying handle, main consider key structure in $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$ as following:

$${G_{ij}}^{\prime}({N_{e}})={\left\{{\begin{array}[]{*{20}{c}}{{p_{i}}}\\ {{q_{j}}}\\ {{N_{e}}}\\ \end{array}}\right\}}(1\leq i,j\leq r=k)$$

Therein, if ${{p}_{i}}$ and ${{q}_{j}}$ is a prime number and satisfy matching requiring of the even established relation of two prime add rum, then ${{N}_{e}}={{p}_{i}}+{{q}_{j}}$ is Goldbach number $G({{N}_{e}})$. We special agreement, if ${{N}_{e}}=2n,n>2$ is any given even, ${{p}_{i}}\in{{N}_{e}}$ is determined prime, ${{q}_{j}}$ is only correspond the matching prime of even established relation of two element add sum, i.e., ${{q}_{j}}={{N}_{e}}-{{p}_{i}}\;(1\leq i,j\leq r)$, then ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ is can called satisfy with pairing requirements of Goldbach number $G({{N}_{e}})$. Otherwise, ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ is called does not satisfy the matching requirements of Goldbach number $G({{N}_{e}})$.

We have obtained new results by the proof as following:

Theorem 3.2. In the model $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$, any given an even number ${{N}_{e}}$, once the element of within the prime set P is determined, by randomly selecting the element ${{q}_{j}}$ within Q,if only at $r\approx 0.8\text{33}\sqrt{{{N}_{e}}}$ computing the number value, then the element ${{q}_{j}}$ in the probability of 50% get one element is the prime, and ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ satisfy the matching requirements of Goldbach’s number $G({{N}_{e}})$.

Proof. Assume ${{N}_{e}}$ be any big enough even number, ${{N}_{e}}=2n,n>2$. In addition assume $N$ as a big enough integer, and ${{N}_{e}}\subseteq N,N=\{1,2,3,\cdots,{{N}_{e}}-1,{{N}_{e}}\}$. In the model $\bmod\,\overset{\equiv}{\mathop{M}}\,({{N}_{e}})$, because of the set $P$ = ${{p}_{1}},{{p}_{2}},\cdots,{{p}_{r}}$ of odd prime in the independent closure of $N$ is may be determined, and the set $Q$ = ${{q}_{1}},{{q}_{2}},...,{{q}_{r}}$ of corresponding odd integers can also be determined uniquely, for $Q\in N,Q=N-P$. Any an element ${{q}_{j}}$ must with another element ${{p}_{i}}$ forming one-to-one mapping to a matching relationship. For ${{p}_{i}}$ be the prime, ${{q}_{j}}$ is either odd or prime. Now it can be considered in this wise, because of all the primes in P are determined, it means that every prime within the r box in $P$ has been selected. What is corresponding the element ${{q}_{j}}$ in $Q$? It is either an odd number or a prime. That is to say, how many is the number of the prime that it is exactly a prime and satisfying the requirements of $G({{N}_{e}})$? We convert directly the problem into a solution ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ whether or not satisfy the matching requirements of Goldbach number $G({{N}_{e}})$ for analysis.

Now might as well assume that $P_{G}^{j}(1\leq j\leq r)$ choose a different scheme from each other that randomly select anyone matching element within ${{q}_{j}}\;(1\leq j\leq r)$ in N, and the result makes ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ all not satisfy the matching requirements of Goldbach number $G({{N}_{e}})$[11]. Here’s appointed: those have been selected element is no longer a part of option again, too.

We firstly consider ${{q}_{1}}$ can arbitrarily select an element in $N$, ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ not satisfy the probability of the requirement of $G({{N}_{e}})$ is ${\scriptstyle{}^{1}/{}_{N}}$. The second let ${{q}_{2}}$ choose another an element in $N$, and ${{q}_{2}}$ $\neq$ ${{q}_{1}}$, and that when selecting ${{q}_{2}}$, ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ not satisfy the probability of the requirement of $G({{N}_{e}})$ is $(1-{\scriptstyle{}^{1}/{}_{N}})$. Third let ${{q}_{\text{3}}}$ choose the remaining elements in $N$, and ${{q}_{\text{3}}}$ $\neq$ (${{q}_{1}}$,${{q}_{2}}$), and that when selecting ${{q}_{3}}$, ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ not satisfy the probability of the requirement of $G({{N}_{e}})$ is $(1-{\scriptstyle{}^{2}/{}_{N}})$,…, and so on. At last, let ${{q}_{\text{r}}}$ choose the remaining elements in $N$, and ${{q}_{\text{r}}}$ $\neq$ (${{q}_{1}}$,${{q}_{2}}$,${{q}_{\text{3}}}$,…,${{q}_{\text{r-1}}}$), and that when selecting ${{q}_{\text{r}}}$, and ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ not satisfy the probability of the requirement of $G({{N}_{e}})$ is $(1-{\scriptstyle{}^{r-1}/{}_{N}})$.

Because ${{q}_{j}}={{N}_{e}}-{{p}_{i}}\;(1\leq i,j\leq r)$ must to determine one ${{q}_{j}}$ element, so ${{q}_{j}}$ within $r$ by random selecting different the prime, either of ${{q}_{j}}$ all not appears it is the prime, ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ not satisfy the matching requirement of Goldbach number $G({{N}_{e}})$; either of ${{q}_{j}}$ even if to appear it is the prime, but ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ still not satisfy the matching requirement of the Goldbach number $G({{N}_{e}})$. Obviously, the probability of thus not satisfied condition that described as follow.

$$(1-\frac{1}{N})(1-\frac{2}{N}){(1-\frac{{\rm{3}}}{N})}{...}(1-\frac{{r-1}}{N})\approx\prod\limits_{i=1}^{r-1}{(1-\frac{i}{N}})$$

Here, we can let $x={}^{i}/{}_{N}$. When $N$ is a large integer, and $x$ is a small real number, according to the following relation of the formula:

$${e^{-x}}=1-x+\frac{{{x^{2}}}}{{2!}}-\frac{{{x^{\rm{3}}}}}{{{\rm{3!}}}}+\cdots$$

Then have $1-x\approx{{e}^{-x}}$. All things considered, the selecting results if $r$ primes does not appear in the $r$ selecting scheme, or ${{q}_{j}}$ even if to appear be prime, but ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ still not satisfy the probability of the matching requirement of Goldbach number $G({{N}_{e}})$, then have the following result:

$$\prod\limits_{i=1}^{r-1}{(1-\frac{i}{N}}){\approx}\prod\limits_{i=1}^{r-1}{{e^{-\frac{i}{N}}}}={e^{-\frac{{r(r-1)}}{{2N}}}}$$

Now, we conversely consider problem can find that, in r selecting scheme, ${{q}_{j}}$ may be selected as the prime more than once in $r$ options, but at least have one prime makes ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ to satisfy the requirements of Goldbach number $G({{N}_{e}})$. Or selecting right only one element is a prime, but it can match ${{G}_{ij}}^{{}^{\prime}}({{N}_{e}})$ which satisfies the requirement of the Goldbach number $G({{N}_{e}})$, the probability of the request as follow.

$${P_{G}}=1-\prod\limits_{i=1}^{r-1}{(1-\frac{i}{N}}){\approx 1-}\prod\limits_{i=1}^{r-1}{{e^{-\frac{i}{N}}}}=1-{e^{-\frac{{r(r-1)}}{{2N}}}}$$

Let’s assume:

$${P_{G}}=\theta=1-{e^{-\frac{{r(r-1)}}{{2N}}}}$$