LiKe Matrix and Goldbach Conjecture

Li Ke

摘 要：This is a briefing.The LiKe Matrix and it's row closure are introduced simply，and on the basis of these deduced Goldbach Conjecture.

關鍵詞：Matrix；number theory；symmetric group；Goldbach Conjecture

基金項目：自然科學夢想基金（WDM2016001）。

Ⅰ.LiKe Matrix

Column of LiKe Matrix is odd number line，and each column begin with prime Numbers，it's specific form see table 1：

From Table 1，we can see that LiKe Matrix have infinite rows and columns；then it can be simply expressed as：the nth column was an odd number line which begin with the nth odd prime number；and the nth column moves down （Pn-3）/2 rows relative to the first column.

For LiKe Matrix，it's easy to see that every column has a prime number（And the column is closed by prime numbers）；So is there at least one prime number in each row？ This is the question about row closure of the LiKe Matrix.

Ⅱ.Row closure's，study

See Table 1，In the odd number line（the first column），exist in Lemma 1：The number of rows in which all the prime numbers are，and the number of the sum of rows number and half the distance of Pn to 3，can constitute a continuous integers.For example，in table 1：In the first column，the numbers of rows which element is prime number are 1，2，3，5，6，8，9，11，14...；The numbers of rows which element is composite number are 4，7，10，12，13...；But the row number of 7 is 3，3add（5-3）/2 equal to 4 and 6+（5-3）/2=7，9+（5-3）/2=10，11+（5-3）/2=12，11+（7-3）/2=13 and so on.This is clearly true in all odd number line，that is the composite numbers in the first column are totally enclosed by the prime number of other columns.Because in the odd number line，the difference value between prime number and prime number just is the half distance of two prime numbers，and the difference value between composite number and prime number pertain to the difference value between prime number and prime number（It's drawer principle：A include 1，2，3，5，6...line number of prime number；B include 1，2，4，5...half from prime number to 3，a∈A，b∈B，it must have c=a+b，c∈C，C include 4，7，10，12...line number of composite numbers），so Lemma 1 is true.These consecutive integers just are row numbers of prime number in all columns.So the row is closed by prime numbers.

Besides，there aren't infinite columns in LiKe Matrix，but LiKe Matrix have Lemma 2：In LiKe Matri，The number of each prime number just equal to the number of odd prime numbers.For example，5 is the second odd prime number，so it Appears twice；7 is the third odd prime number，so it Appears three times.And all prime numbers are like this，this ensures that all the row number of prime number just can add all half distance between prime number and 3，that is every line has enough number of columns to ensure prime number can closed this row.Q.E.D！

Ⅲ.Inference（Goldbach Conjecture）

What's so magical about LiKe Matrix？ In particular，each row has a prime number.Scrutinize table 1，for each row，you can figure out every number by the first odd number and the prime distance.When each row has a prime number，there is the following formula：

O-（Pn-3） = p （1）

O is the first odd number of row（All the odd-numbers）；Pn is the first prime number of the specific column；and p is a prime number.

The formula （1） is equivalent to O+3=p+Pn，O+3 is the integer，and it's all the integers greater than or equal to 6.Is this the expression of Goldbach conjecture？So LiKe Matrix is a mysterious and strange special matrix，as for it's other secrets，it remains to be seen.

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