Brief: A new method to understand the Goldbach conjecture – LiKe Sequence

The strong Goldbach conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. It seems simple, but it has not been solved since 1742^[1,2]. So the current approach is not very promising. The purpose of this paper is to propose a new method, this method is simple and easy to understand, and it’s useful to solve the problem by conventional methods. This paper only makes a brief statement, only hoping to be helpful to the study of Goldbach conjecture.

1. New understanding about Goldbach conjecture

Goldbach conjecture says every even integer greater than 2 can be expressed as the sum of two primes. Of course, every even integer greater(2N) than 2 can be expressed as the sum of one prime(≤N, except 2) and another integer. Please see the table1 as below.

Table 1. The relationship between even integer and primes

As seen form the table 1, for every even integer greater than 2, in it’s corresponding integers(sequence), have a prime number. if not, Goldbach conjecture will be wrong. In addition, we can alse see that, The factor of each composite number can only be a prime number up to N.

2. The ideal of sequence

Now, Let's take a look at a new sequence, please see the table2.

Table 2. Odd primes and their LiKe sequence

From the table 2, we can see that, For given odd primes, the product of these can be used to represent an infinite number of composite numbers(almost composite numbers). If some numbers are spaced the same as prime numbers in reverse order, called them LiKe sequence. such as 9 or 27 are the LiKe sequence of 3 and (25,27) is a LiKe sequence of (3,5) in the table 2. That is to say: in the table 1, if all of 43,49,51,57,61,63,67,69,73,75,77 are composite numbers and the factorization can only contain 3,5,7,11,13,17,19,23,29,31,37, they will be the LiKe sequence of (3,5,7,11,13,17,19,23,29,31,37).

3. Proof line

So, Just have to prove that when P_n is big enough, there's no LiKe sequence for the odd prime sequence (3,5,7,...,P_n), or that the LiKe sequence has to be greater than 2P_n+1. It doesn't look too hard, at least, when P_n is big enough, the LiKe sequence of (3,5,7,...,P_n) must be greater than 3P_n, and 3P_n almost more than 2P_n+1. So this is a problem that can be solved with modern mathematics.

4. Summary

All of above are obvious, and if they are true, we will get: (3,5,7,...,P_n) has no LiKe sequence before 2P_n+1. It means that the Goldbach conjecture is true.

References:

[1]. Chen, J. R. ON THE REPRESENTATION OF A LARGE EVEN INTEGER AS THE SUM OF A PRIME AND THE PRODUCT OF AT MOST TWO PRIMES[J]. Science China Mathematics, 1978, 21(4):421-430.

[2]. Heath-Brown D R , Puchta J C . Integers Represented as a Sum of Primes and Powers of Two[J]. Asian Journal of Mathematics, 2002, 6(3):535-565.