Rigorous Proof of Goldbach’s Conjecture

In this article, we use set, function, sieve and number theory to study the prime and composite numbers, prove that the lower limit formula of the number of prime numbers derived from the Euler’s function, and find d(n) to count the lower limit formula of the number of prime integer-pairs. We proved that Goldbach’s conjecture is correct by mathematical induction. Finally, we proved proof reliance by mathematical analysis and computer data.


Journal of Applied Mathematics and Physics
jecture is equivalent to E(x) = 1 [3]. We will use the method of exception set to solve the Goldbach's conjecture.

Remarks on Notation
Definition: If A is a set, card (A) is the number of elements of set A. In addition:  n p A is the set of integers with multiples of prime p. As  π(N) is the number of prime numbers in [1, 2n].  D(N) is the number of prime integer-pairs that sum of two integers is equal to 2n.
 D n is a set with elements are prime integer-pairs.

Prime and Composite Numbers
In number theory, we know the number of primes can calculate by sieve of Eratosthenes and inclusion exclusion formula [3] [5] [6].
It can be received exact value by the calculation of the tolerance formula, but it can only be used in small range of positive integers, because the operation is very complicated. For a larger range of data operations, there is inefficiency.

Function ϕ(n)
Theorem. (Euler function) [3] [5] [6] If n contains any prime number p i , p j , •••, p k factor, let ϕ(n) be the number of positive integers not greater than and prime to n, then ( )  [1, n]. Then we take ϕ'(n) from ϕ(n) as below ( ) By Theorem 60 in "An Introduction to the Theory of numbers" [5], we known ϕ'(n) is multiplicative. According to Theorem 3, it can be known that (1 − 1/p) is the eliminating factor of the composite number. By Theorem 3, the every eliminating factor (1 − 1/p) less than or equal to the actual value, then its means that, ϕ'(n) is lower bound of the number of primes in [ Proof. By Theorem 4, we known, if n is a multiple of primes 2, 3, •••, p m , But when n is not a multiple of primes p i , there maybe has a error r In order to obtain a definite lower limit of the number of primes and simple operation, we find a function that may be associated with the error as an error compensation. That is to subtract 1 is the error compensation for every opera-

Integer Pairs
2n positive integers be arranged to w rows as Figure 1 follows.
We obtain n pairs of two integers that sum are just equal to 2n, it is called "integer-pair". It is called "prime integer-pair" if two integers are primes.
Obviously, as long as we can prove that there has one or more than one prime integer-pair, the Goldbach's conjecture problem can be solved [7]. Theorem 6. Let D n be a set of prime integer-pairs in Figure 1. Then Proof. First, to delete all even integer-pairs in Figure 1, we obtain n/2 (or (n + 1/2)) odd integer-pairs, as Figure 2 (assume n − 1 is odd) noticed that the spacing of each odd prime number is the same as that of the natural number, as shown in Figure 3. So the function formula of ϕ'(n) can also be applied (except without the 2 factor).
Our approach is: 1) First, those integer-pairs on the top row that contain multiple of 2, 3, •••, p m are to marked "/", (suppose n − 3 are composite numbers) (see Figure 4). Journal of Applied Mathematics and Physics by Theorem 5.
2) Next, those integer-pairs on the bottom row that contain multiple of 2, 3, •••, p m are also to marked "\", see Figure 5 (suppose n − 3, n + 3, 2n − 5 are composite numbers); Then, we are deleting all integer-pairs that are marked "/" or "\". At same times, we are using function d(n) for calculation the number of all deleted integer-pairs as below(by change every numerator 1 to 2 from function ϕ'(n)) ( ) ( ) In this way, we counted only one time for every integer-pair that containing one multiple of 2, 3, •••, p m ; but for other integer-pairs that contained two multiples of 2, 3, •••, p m (as integer-pairs noted "×"), we repeat counted it by two times, this is called excessive duplicate sieving [8]. Note: this repetitive deletion will make d(n) − m less than the actual number of the remainder with 1 and prime numbers (see Figure 5). This does not affect the validity of the proof. Instead, the way to do this is to get a more rigorous the lower limit of card (D n ) (i.e. D(N)), which is exactly what we need. So, the(4) is true.  Proof. Because all integer-pairs that containing of the composite numbers have been deleted, the remnants are just 1 and primes. Because the D n is a set of prime integer-pairs in n integer-pairs, and ( ) In order to get the safe and reliable data, our rule to believe that if d(n) is greater than or equal to 2, the card (D n ) is greater than or equal to 1 (because 1 and three primes can form two integer-pair). So Theorem 7 is true. □ 2) Suppose when m = k,   Namely there is having one or more one of prime integer-pairs exist in n integer-pairs that sum of two primes equals 2n, witch when 60 m ≥ ,

Safety Analysis
Some people may ask: is the method of repeated screening reliable? Is it beyond the range of positive integers?
Now let's do a mathematical analysis of this problem [9] [10] [11]. greater than 1 if p m (namely 2n) is enough big. There is also has a space of prime integer-pairs. In other words, the data is safe and reliable. Such as, the safety feasibility of repeated screening method is proved by the mathematical analysis [8] [9]. In addition, Here we see that the function d(n) increases as p m increases, and d(n) is a monotonic increasing function.

Journal of Applied Mathematics and Physics
There is a large surplus for the proof of Goldbach's conjecture, as shown in: 1) By computer calculation, the actual number of prime pairs is less than d(n) − 1. That is to say, the error compensation is not need to minus m. So, when m ≥ 5, p m ≥ 11, p m ≥ 2.75 > 2, the Proposition 4.1 can be proved.
So the goldbach's conjecture is correct absolutely.

Computer Test Data
Through computer programming, we get Table 1 as follows.
1) Table 1 shows that ϕ'(n) is less than π(N). This shown the ϕ'(n) is the lower limit of the number of prime numbers.
2) Table 1 shows that d(n) is less than D(N)and correct. The general trend of d(n) is upward ; with the increasing of 2n, the number of prime integer-pairs is increasing. In other words, even number N (=2n) goes to infinity, the Goldbach's conjecture is still correct.
By computer programming, we obtain Figure 9. Those digits represents the number of prime integer-pairs in [150, 300], (the detection interval is 2); 2) the dotted line above represents the function d(n)，and below is d(n) − m. The solid line shows the p m /4.  Figure 9. Diagram of the number of prime integer-pairs. Journal of Applied Mathematics and Physics Figure 9 is showing: 1) All digital with the number of prime integer-pairs above the line of d(n) − m, which more and more. 2) This is to explain that the function d(n) is the lower bound of the number of prime integer-pairs, p m /4 and d(n) − m are its the reliable lower limit. That means that, when x ≥ 100, the number of prime integer-pairs must be greater than 1. In other words, no matter how big x is, E(x) is always equal to 1.

Conclusions
Through the above proof, we proved that even x greater than 2 can find prime pairs that sum of two primes equal to even x, which proves that the exception set E(x) is always equal to 1. In this way, we proved the Goldbach's conjecture by the method of exception set [3].
In addition, it is worth mentioning that this proving method is equally applicable to solving the same types of problems, such as Angle valley conjecture and a proof of the infinite of twin prime numbers (let every integer-pair be that the difference of two integers is 2). A world mathematical problem unsolved for more than 270 year was finally solved successfully. I have to say that this is a gratifying success case of mathematical theory and computer technology.