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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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    $124=13+17+19+23+23+29$. There was no statement that the primes are distinct. – 2017-02-10
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    Yes, $124=113+11$, which is sum of two primes. The Goldbach conjecture claims this can be done for any even number. – 2017-02-10
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    "There was no statement that the primes are distinct" Thanks – 2017-02-11
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    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. – 2017-02-11
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    Ok, but how use this "multiples" to arrive in something like this: 7 + 19 + 53 + 73 = 152 – 2017-02-11

3 Answers 3

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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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    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? – 2017-02-10
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    @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. – 2017-02-10
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    "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. – 2017-02-10
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    @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 ($i=6$). – 2017-02-11
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    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) – 2017-02-11
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    @DiogoAlcântara I understand that. What I'm saying is: *why do you think your conjecture implies Goldbach*? You claimed that the two are equivalent, but have given zero justification for it. – 2017-02-11
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    That is: I'm not arguing against your conjecture, I'm arguing against **your claim that it is equivalent to Goldbach**. – 2017-02-11
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    My justification is the same as we learning in preschool. Even with Even -> Equal nature - > Generates Even. Odd with Odd -> Equal nature -> Generates Even. Even with Odd -> Different nature -> Generates Odd. Odd with Even ->Different nature -> Generates Odd Since all prime numbers are odd (With the exception of 2,), the rule is the same. – 2017-02-11
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    This nature equivalent in between, quantity and value, can not generate something of different nature. Amount even generates even, Amount odd generates odd. However, my proof, Is not proof, is something more logical and philosophical than mathematics. About Goldbach Conjecture, he referred only about sum of 2 number. (Which also is an amount even) – 2017-02-11
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    @DiogoAlcântara "My proof, Is not proof, is something more logical and philosophical than mathematics." No, it's not more logical; what you've written is basically nonsense. Even and odd numbers interact in complicated ways; there is no general principle of the kind you're trying to express. Regardless, I'll say again: **you haven't argued that your conjecture implies Goldbach!** Your observation "Which also is an amount even" shows that *Goldbach implies your conjecture*, like I said above. But the converse is not something you've given any argument for. What you've written isn't mathematics. – 2017-02-11
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    I understand. The only similarity is the quantity of numbers in the sum, needed for generate a even number: 2,4,6,8.etc. Now, I think it is impossible for two even numbers to generate an odd number. Do you know any examples that proves the opposite of my "general principle"? – 2017-02-11
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    @DiogoAlcântara Well, the problem is that that general principle is so vague, I don't even know what it means. "I think it is impossible for two even numbers to generate an odd number" - what does "generate" mean? Obviously there are lots of ways to combine two even numbers and get an odd number (silly example: division (${6\over 2}=3$)). Before one can try to prove or disprove something, it must be made **precise**. – 2017-02-11
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    Ok. My native language is Portuguese. The challenge: Show me a single example where: the sum of a "quantity even" of "odd numbers" be equals an "odd value". Example: 1+1=2 (quantity even, value even) Example: 1+3=4 (quantity even, value even) Example: 1+3+5+7=16 (quantity even, value even) If you find it, then my "general principle" is vague and nonsense. Otherwise it is not. – 2017-02-11
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    @DiogoAlcântara Ah. This is now precise, and in fact easy to prove: anytime you add up an even number of odd numbers, you get an even number (this is a good exercise). In particular, anytime you add up an even number of odd *primes*, you get an even number. However, this **does not** prove the statement you're interested in! You're trying to go the *other way*: start with an even number, and write it as a sum of even-many primes. (cont'd) – 2017-02-11
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    Here's the mistake you're making, more abstractly: you know "Every $A$ is a $B$" - in this case, "every [sum of an even number of odd primes] is a [even number]". But what you want is different: you want "Every $B$ is an $A$" - that is, "every [even number] is a [sum of an even number of odd primes]". In general, "Every $A$ is a $B$" *does not imply* "Every $B$ is an $A$", so your (easily proved) "general principle" doesn't have anything to do with the conjecture you're interested in. – 2017-02-11
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    Ok. I need to study it all more calmly. I was researching about "IChing" and "problem solving" and I ended up in this theorem nothing related with the theme initial. Hehe Thanks for the answers it was fun. ^ ^ – 2017-02-11
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    I arrived at this: number of possible combinations in between two odd numbers to form a even. Equation: C=n/2(+1 is this is odd)/2. Where: n = even number; C = number of possible combinations in between two odd numbers to form a even. I put the examples in the answer below. – 2017-02-11
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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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    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. – 2017-02-11
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    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. – 2017-02-11
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    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. – 2017-02-11
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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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    Dislike without exposing the motive is just envy. Sorry. – 2017-02-11