Goldbach Conjecture and Conjecture of Preservation of Nature of Numbers

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About Goldbach Conjecture: https://en.wikipedia.org/wiki/Goldbach%27s_conjecture

Doubt 1

"The best known result is due to Olivier Ramaré, who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most six primes.".

About this legation I think: Or, this is wrong. Or, this is badly formulated.

The number 124, for example, is the sum of 8 prime numbers. 5+7+11+13+17+19+23+29 = 124

That is, it has already exceeded that maximum sum of 6 prime numbers.

How is this interpreted?

Doubt 2

Goldbach conjecture

"Every even integer greater than 2 can be expressed as the sum of two primes".

I concluded that this conjecture is equivalent:

"Every EVEN integer greater than 2 can be expressed as the sum of an amount EVEN of prime numbers".

That is, 2,4,6,8, etc.

I published my thoughts here

http://psicolagem.blogspot.com.br/2017/02/goldbach-conjecture-2017-or-conjecture.html

Can you see something wrong with that?

3

There are 3 best solutions below

17
On

For your first question, you are misunderstanding the statement. Ramare proved that if $n$ is an even number $\ge 4$, then we can find prime numbers $p_1, p_2, . . ., p_i$ for some $i\le 6$ such that $n=p_1+...+p_i$. That is, for each even $n\ge 4$ there is some $i\le 6$ such that $n$ can be written as the sum of $i$ primes.

For example, you look at $124$; well, $124$ can be written as the sum of two primes ($113+11$), and two is at most (that is, $\le$) six. The fact that such an $n$ can also be written as the sum of more than $6$ primes, has nothing to do with Ramare's result.


For your second question, you give no justification at all: how is it that you think Goldbach is equivalent to your statement? Certainly Goldbach implies it since $2$ is even, but how on earth do you claim that the converse holds? Suppose you could write an even $n\ge 4$ as the sum of $24$ primes ($24$ is just some random even number); how would you use this to write $n$ as the sum of $2$ primes?

3
On

Here's what I think, I hope it helps:

Doubt 1: and $20=2+2+2+2+2+2+2+2+2+2$. The result state that every even number can be written as the sum of six primes at most.

But I don't think that this is the best approximation to Goldbach Conjecture. I recommend you to look for the Weak Goldbach conjecture, which is already proved.

Doubt 2: It is obvious that Goldbach Conjecture implies your conjecture, but the other way it is not so clear. But for to declare that the other implication is not true with a counter example we must find an even number which can be expressed as the sum of 4 primes but not as the sum of 2 primes; and this is equivalent to deny Goldbach conjecture, which it is believed right.

1
On

About Doubt 2:

I arrived at this: number of possible combinations in between two odd numbers to form a even. Equation: C=n/2(+1 if this is odd)/2

Where:

n = even number

C = number of possible combinations in between two odd numbers to form a even

Example: 14

C=14/2=7(is odd then)+1=8/2=4

14 is formed by 4 possible combinations in between two odd numbers

(1) 13+1

(2) 11+3

(3) 9+5

(4) 7+7

Example: 24

C=24/2=12(is even)/2=6

24 is formed by 6 possible combinations in between two odd numbers

(1) 23+1

(2) 21+3

(3) 19+5

(4) 17+7

(5) 15+9

(6) 13+11

Example: 54

C=54/2=27(is odd)+1=28/2=14

54 is formed by 14 possible combinations in between two odd numbers