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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?

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  • $\begingroup$ $124=13+17+19+23+23+29$. There was no statement that the primes are distinct. $\endgroup$ – Thomas Andrews Feb 10 '17 at 23:20
  • $\begingroup$ Yes, $124=113+11$, which is sum of two primes. The Goldbach conjecture claims this can be done for any even number. $\endgroup$ – Wolfram Feb 10 '17 at 23:23
  • $\begingroup$ "There was no statement that the primes are distinct" Thanks $\endgroup$ – Diogo Alcântara Feb 11 '17 at 0:51
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    $\begingroup$ Your conjecture in 'doubt 2' is, in fact, trivial. Every even number is either $2+2+2+2+\ldots$ with an even number of terms (if it's a multiple of four) or $3+3+2+2+\ldots$ with an even number of terms (if it's two more than a multiple of four). So while it could 'imply' Goldbach (if Goldbach is true) it will be no easier to prove this than to prove the Goldbach conjecture itself. $\endgroup$ – Steven Stadnicki Feb 11 '17 at 5:02
  • $\begingroup$ Ok, but how use this "multiples" to arrive in something like this: 7 + 19 + 53 + 73 = 152 $\endgroup$ – Diogo Alcântara Feb 11 '17 at 5:14
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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?

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  • $\begingroup$ So, does it mean that sum greater than 6 can generate even number, but it he has not proven. He has proved only with at most 6? $\endgroup$ – Diogo Alcântara Feb 10 '17 at 23:43
  • $\begingroup$ @DiogoAlcântara I don't quite understand your comment, but: what Ramare proved is the following. If $n\ge 4$ is even, then we may write it as the sum of at-most-$6$ primes. We may also write it some other way; but the point is that Ramare showed that we can write any number as a sum of primes reasonably efficiently, that is, using few (in this case, $\le 6$) primes. Goldbach is a strengthening of this: it states that in fact you never need more than $2$ primes. $\endgroup$ – Noah Schweber Feb 10 '17 at 23:45
  • $\begingroup$ "We may also write it some other way". That's what I understood: we can write in other ways, but he only proved the sum of at most 6. $\endgroup$ – Diogo Alcântara Feb 10 '17 at 23:52
  • $\begingroup$ @DiogoAlcântara In that case, yes. But I object to the phrasing "he only proved . . ." Remember, as $i$ gets smaller, proving "Any even $n\ge 4$ can be written as the sum of $\le i$ many primes" gets harder: the smaller the $i$, the more impressive the result. Goldbach is the ultimate ($i=2$), and what Ramare showed was a weaker result along the same lines ($i=6$). $\endgroup$ – Noah Schweber Feb 11 '17 at 0:02
  • $\begingroup$ Ok. About the second question: I do not claim that 24 is the sum of 2 cousins. I claim that every even number is the sum of an amount even of prime numbers. example: 26= 3+5+7+11 (4 numbers) | 26 = 7+19(2 numbers) $\endgroup$ – Diogo Alcântara Feb 11 '17 at 0:05
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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.

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  • $\begingroup$ What I've gotten is just realizing that a even number is a result of an amount even of prime numbers: it can be 2,4,6,8. Example: 26= 3+5+7+11 (4 numbers) | 26 = 7+19(2 numbers). And the reason for this, I wrote in the article there, is due to the nature equivalent in between, quantity and value, which can not generate something different. $\endgroup$ – Diogo Alcântara Feb 11 '17 at 0:14
  • $\begingroup$ I understand (I just put 4 above as an example). My point is that the step of proving that your conjecture implies Goldbach's conjecture (which is essential to have an equivalence) doesn't seem right. For example, proving this implication is the same as to prove: if the Goldbach Conjecture isn't true, i.e., there exist an even number that can't be written as the sum of two primes numbers, then that number can't be expressed as an even amount of prime numbers. But this statement does not 'feel' right. Without proof, one can only conjecture that both conjectures are equivalent. $\endgroup$ – Fernando Feb 11 '17 at 1:04
  • $\begingroup$ I understand. The only similarity is the quantity of numbers in the sum needed for generate a even number: 2 or 4,6,8.etc. That is, if someone prove that exist an even number that can't be written as the sum of two primes numbers, this my conjecture is still valid, and takes the place of the previous one. Because, the even number can not be written by 2, but maybe it can by 4, 6, 8, etc. However, I agree with Euler: I'm sure the Goldbach's conjecture is correct, but I can not prove it. $\endgroup$ – Diogo Alcântara Feb 11 '17 at 1:32
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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

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  • $\begingroup$ Dislike without exposing the motive is just envy. Sorry. $\endgroup$ – Diogo Alcântara Feb 11 '17 at 5:00

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