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1.
I define a distinct sum as the addition of two primes such as 3+5, 3+7, etc. The addition of two identical primes is not a distinct sum. I define presentations of an even number as the number of distinct sums this number is equal to eg. 1+19=20, 3+17=20, 7+13=20, we say the even number 20 has three presentations. I define a column as a sequence of distinct sums containing each odd prime to which is added all primes greater than this prime in a given quantity 10^n where n a non zero natural number. For example, for n=1, 1+3, 1+5, 1+7 and 3+5, 3+7, are two columns within 10^1. I define a map as the number of columns which are contained within a quantity 10^n. For example, for n=2 we have 24 odd primes and the map 10^2 contains 24 columns; the first column comprises the odd number 1 to which is added all odd primes within 10^2. Maps can be calculated vertically or horizontally. In this new approach all calculations are based in the vertical sequence of distinct sums. Let’s make a map within the quantity 10^2 so we obtain:
1+3
1+5 3+5
1+7 3+7 5+7
1+11 3+11 5+11 …
1+13 3+13 5+13 …
1+17 3+17 5+17 …
. . . …
. . . …
. . . …................... 79+83
1+89 3+89 5+89 …....79+83 83+89
1+97 3+97 5+97 …... 79+97 83+97 89+97
The columns of the map contain numbers of presentations in the following sequences:
24, 23, 22, … 3, 2, 1.
Because of this structure of sequences we can calculate the number of presentations within the quantity 10^2 with the formula (f+1)f/2 where f is the number of odd primes>1. And so we have: Number of presentations=(24+1)24/2=300 within 10^2. Because we want to exclude all presentations that are greater than 10^2, for the time being we will accept that half of the number of presentations is ≤100, so Number of presentations 10^2=300/2=150.
The number of even numbers contained in the quantity 10^2, excluding 2, is 49. So for each even number contained in 10^2 we obtain almost three presentations. This does not mean that each even number has three presentations. From the prime number theorem we have 10^n/ln10^n≈f primes so
(f+1)f/2≈10^2n/2(ln(10^n))^2
If n=2 then 10^4/2 (ln100)^2≈236. This result is a close approximation of the actual number of presentations within 10^2, which is 300. The number of even numbers contained in the quantity 10^n, excluding 2, is obtained: Even numbers=(10^n/2)-1 and so we obtain
[10^2n/2[ln(10^n)^2]/[((10^n)/2)-1]=10^2n/[2(ln(10^n)^2)(10^n-2)]
Because we have accepted only half of total presentations are equal or less than 10^n the last result is written 10^2n/[2(ln(10^n))^2(10^n-2)]. The actual number of presentations in the quantity 10^2 whose distinct sums exceed 10^2 is 104 and the actual ratio is (300-104)/49=4 which means within the quantity 10^2 we have 4 presentations for each even number. From the last relation we obtain 10^4/[2((ln100)^2)98]≈2.4 which means 2.4 presentations within 10^2 for each even number. The numerical value of the ratio 10^2n/[2(ln(10^n)^2)(10^n-2)] tends to infinity as n tends to infinity.

All even numbers greater than two can be written as 2m+2 and when m takes values from one to infinity all even numbers are presented. Also the sum of two prime numbers is written as 2k+2. If we prove k takes all values m takes then the Goldbach conjecture is proved.
When p_1=4x+1 and p_2=4x_1+1 then m=[(2x+1)^2+(2x_1+1)^2-2]/2-2x^2-2(x_1)^2
When p_1=4x+1 and p_2=4x_1+3 then m=[(2x+1)^2-(2x_1+1)^2-2]/2+(2x+2)^2-2(x_1)^2
When p_1=4x+3 and p_2=4x_1+3 then m=(2x+2)^2/2+(2x_1+2)^2/2-[(2x+1)^2+(2x_1+1)^2-2]/2.
These three cases cover all the sums of prime numbers. That means k takes as many values as the number of presentations in a given quantity 10^n.
So (k number of values)/(m number of values) tends to infinity as n tends to infinity.
From the above we can show by shear calculation m takes all values k takes from 1 to 100 and from 100 to 1,000,000 and from 10^6 to 10^9 and because the relation of k to m is asymptotic, m will take all values k takes.
My question is, is this form of mathematical induction correct or incorrect? If incorrect, what are the logical arguments to support such an opinion?
If this question is not acceptable as it is, please suggest improvements.
• CommentAuthorMariano
• CommentTimeJan 13th 2012 edited

how can I apply mathematical induction to prove k takes all the values m takes, having in mind that no one can prove that k does not take all values of m?

The form if this question is «How can I prove X, having in mind that no one can prove the negation of X?»

If I understand correctly, the answer is: no one knows...

2.
I think it is ok know to post my question on MO.
3.
How is

"I define a column as a sequence of distinct sums containing each odd prime to which is added
all primes greater than this prime in a given quantity 10^n where n a non zero natural number"

supposed to be parsed?
4.
Thanks for the suggestions. I made the chances.
5.

You have the sentence, "If we prove k takes all values m takes then the Goldbach conjecture is proved." Then, you have the question "From the above facts, how can I apply mathematical induction to prove k takes all the values m takes?"

In other words, you are asking us how to apply mathematical induction to prove the Goldbach conjecture. This is not an acceptable question. What sort of answer are you expecting?

6.
Dear Scott.
I do not want you to apply mathematical induction. What I am asking is whether, from the facts presented, mathematical induction is applicable. If it is true for the first 100 even numbers and then to the first million or billion it should be true for any zillion even numbers. The facts are proving the relation of K and M is asymptotic. I hope this alleviates your concerns about the intent of my question.
"What sort of answer are you expecting?" I want to know if there is any flaw in my way of thinking.
Vassili
• CommentAuthorMariano
• CommentTimeJan 14th 2012

Ah well, if it should be true, then...

7.
Dear mariano.
I edited the question and i will appreciate any comments .
8.

I can answer your question here. A large collection of examples is not a proof, unless it is large enough to be exhaustive. Here is a MathOverflow big-list question that contains lots of promising patterns with eventual counterexamples. None of those cases admit proofs by induction, even with zillions of affirmative examples, because the general claims are false. In the case under consideration, you cannot expect a proof by induction to appear without first revealing more fundamental structure in the primes. In particular, probabilistic arguments using the prime number theorem need to be supplemented with strong error bounds.

9.
Dear Scott.
Thank you for the information and your opinion. I want to bring to your attention to the following. We can measure horizontally the even numbers on each map. If we do that in the following way we always obtain a number of even numbers which do not appear again. If p_1 is the immediate prime smaller than (10^2)/2 and if p_2 is the immediate prime less than 10^2 then all even numbers equal to or smaller than (10^2)/2 do not appear after p_2.This process is an exhaustive process containing in each step a greater number of even numbers which exist within a bound set by the immediate prime numbers as above. Because of the gravity of the problem I think it is fair to be posted so others will have the opportunity to express an opinion.
• CommentAuthorMariano
• CommentTimeJan 15th 2012

What you want to ask is «Can I use this idea to prove the Goldbach conjecture?».

This is not a good question for MO. This has already been stated clearly by Scott, who has even answered your question in the only possible way.

Maybe this thread should be closed?

10.
Dear Mariano you and Scott you are entitled to your opinions without presenting any mathematical arguments. Every time I try to make my arguments as clear as possible, backing them with logical facts. If anyone brings a sound mathematical argument which contradicts my ideas I will accept it. If you want to force your opinions, I have no way of stopping you.
• CommentAuthorYemon Choi
• CommentTimeJan 15th 2012

Vassilis, Mariano and Scott are not alone in their opinions. While you may view the calculations you have given as logical facts or evidence, they do not amount to a proof; and moreover, asking people if there is a way to turn them into a proof is actually, from the point of view of mathematics, asking them to do all the work.

You say, originally

From the above we can show by shear calculation m takes all values k takes from 1 to 100 and from 100 to 1,000,000 and from 10^6 to 10^9 and because the relation of k to m is asymptotic, m will take all values k takes.

This is not proof, this is observation followed by an assertion ("m will take all values k takes").

My question is, is this form of mathematical induction correct or incorrect?

Incorrect.

If incorrect, what are the logical arguments to support such an opinion?

Do you mean, the logical arguments to support the opinion that this is incorrect? The burden of proof is not on us; the burden of proof is on you. What you have done may be, in your opinion, overwhelming evidence, but it is not a rigorous mathematical argument. As Scott has already said above, mathematical induction is not just a matter of observing many examples and saying "well, surely it must be true". If you dispute this basic premise, then I am sorry but it seems unlikely that there can be worthwhile engagement between you and this community.

If you post this question on MO, I will vote to close it. I also think that the present thread should be closed.

11.

I agree with Yemon's last paragraph, and I can safely say that any question on possible proofs to Goldbach using the techniques outlined will be closed, and quickly. This is not an opinion, but borne out by experience with a) mathematics b) mathematicians and c) the rules of this site.

12.
Vassilis Parassidis -- I really don't know what you hope to accomplish with your repeated posts to meta. It should be clear to you by now that these kinds of questions are not welcome on MO and none of the active participants here are going to engage with you. Frankly, I'm surprised that the moderators have not simply asked you to leave entirely (like they have done with other people in similar situations). You are wasting your own time here.

I also want to add my voice to the chorus of people asking for this thread to be closed.
13.
Dear Yemon,
Scott has asked me to present an exhaustive process which i did as you can see above. With this process we obtain the asymptotic relation as presented above. Since I received three threats you can go ahead and close the thread since you consider yourselves the only experts.
14.