Goldbach's weak conjecture

Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum.) This is called Goldbach's weak conjecture. I think that it is almost the same as Goldbach's conjecture.

a＞2
2a+1=P1+P2+P3
2a=(P1+P2+P3)-1

P is prime numbers.

a＞2
2a+1=O1+O2+O3
2a=(O1+O2+O3)-1

O is odd numbers.



For example,

2a=6 (a=3)

6=(3+2+2)-1 This is prime numbers.

6=(3+3+1)-1=(5+1+1)-1 This is odd numbers.


2a=8 (a=4)

8=(5+2+2)-1=(3+3+3)-1 This is prime numbers.

8=(7+1+1)-1=(5+3+1)-1=(3+3+3)-1 This is odd numbers.


2a=10 (a=5)

10=(7+2+2)-1=(5+3+3)-1 This is prime numbers.

10=(5+5+1)-1=(7+3+1)-1=(9+1+1)-1=(5+3+3)-1 This is odd numbers.

∴ (P1+P2+P3)-1⊂(O1+O2+O3)-1⊂2a

Therefore, Goldbach's weak conjecture is correct.

Goldbach's conjecture

Every even integer greater than 2 can be expressed as the sum of two primes. This is called Goldbach's conjecture.

2a=P1+P2

when a is 2, 4=2+2.

In the case of a>2, P1 and P2 must be odd numbers.


P is prime numbers. O is odd numbers.

∴ P1+P2⊂O1+O2⊂2a

I have found the pattern of prime numbers which is based on odd numbers.

P1+P2⊂O1+O2⊂2a is never broken.
It is quite natural that you can keep finding P1.
P2=2a-P1



Therefore, Goldbach's conjecture must be correct.