Observation to Goldbach's conjecture

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every evennatural number greater than 2 is the sum of two prime numbers.

If we align the numbers in this 6-column form, we can find some rules for summing the primes to give an even number in the columns with even numbers.

Let's see the column with numbers (4,10, 16 ... ) now; let's call it column #4; the conjecture says every even number n = p1+p2, even those numbers which are in column #4.

n%6 = p1%6 + p2%6, in this column we know that the n%6=4 for every numbers in this column. We have primes mostly in Column #1 and Column #5.

Let's check the number 10: 10 = 7+3 = 5+5.

if n = p1 +p2 and we selected p1 from the first column, that means p1%6 =1, then the second prime number should satisfy p2%6=3.

if n = p1 +p2 and we selected p1 from the fifth column, that means p1%6 =5, then the second prime number should satisfy: p2%6=5, and there is only one single prime where p%6=3 is true that is the number three.

So, we can be 100% sure that every number in the 4th column can be built up in two ways:

1. n= 3 + p2, where p2%6=1

2. n = p1 + p2, where p1%6 =p2%6 = 5.

We can make similar statements to the 2nd and 6th columns as well...

1st case:

Further, we need to notice if we have a prime number in the 5th column, it means that we can build up all of the even numbers based on this prime:

e.g. : let it be the selected prime 29; in the next row we have three even numbers: 32, 34, 36.

29+3=32

29+5=34

29 +7=36

2nd case: if we don't have prime in the 5th column in a row r. Then, if we have a prime in the first column in the row r and r+1, it means we can build up the even numbers with the sum of two primes again. e.g.: 35 is not a prime, but in the first column in the same row, 31 is a prime and 37 is a prime as well, which means the 38, 40, and 42 can be built up like this:

38 = 31 +7

40 = 37 +3

42 = 37 + 5