I did some research and I could find anything similar to what I am going to attempt to describe. What I would like to know if it is really useful or if I am wrong and someone has already described mathematically similar.
Just to give some background. Earlier this year I came up with a proof that describes the relationship between primes greater than 3 and their relationship. In essence what the proof is saying is that depeding on the difference between two primes they can be expressed in different ways. In fact they are 4 groups that describe all the potential differences between these two primes. This is proven and if you are in doubt I will e-mail it to you.
What I then decided to do was to add these equations that describe all the possible primes. It isn’t as simple as 6n+1 and 6n-1, although the equations are equivalent. Mine are a little more in depth. When I did this I ended up with six equations.
12n
12+2
12n+4
12n+6
12n+8
12n+10
This would describe all evens greater than 12. This doesn’t prove Goldbach’s conhecture obviously. But does it provide a unique and useful way of looking at it?
To have your observation be anything that’s not completely trivial, you would have to say something about each of those categories. You would have to say something like this:
All even numbers can be expressed as one of the following six forms for a positive whole number n:
12n
12n + 2
12n + 4
12n + 6
12n + 8
12n + 10
But we know that numbers of the form 12n can be expressed as AAAAA.
And numbers of the form 12n + 2 can be expressed as BBBBB.
And numbers of the form 12n + 4 can be expressed as CCCCC.
And numbers of the form 12n + 6 can be expressed as DDDDD.
And numbers of the form 12n + 8 can be expressed as EEEEE.
And numbers of the form 12n + 10 can be expressed as FFFFF.
You would have to fill in AAAAA, . . . , FFFFF so that you say something about that category. You didn’t, which means that so far you’ve only made the most trivial observation.
BTW, the weak Goldbach conjecture() has apparently been provenunconditionally very recently. ( - every odd number greater than 5 is the sum of three primes.)
I think what you’re trying to get at is that all primes being of the form 6n-1 or 6n+1, together with your way of describing all even numbers in terms of their modulus with respect to 12, could be the way to prove Goldbach’s conjecture that all even numbers > 4 are the sum of two primes.
HOWEVER. It seems to me the big sticking point is that Euler didn’t say, because it isn’t true, that every number of the form 6n-1 and 6n+1 are primes.
“if prime, then 6n+1 or 6n-1”.
is not the same as “if 6n+1, or 6n-1, then prime.”
So you could look at your six categories (let’s call them Mod0, Mod2, Mod4, etc.) and demonstrate that each can be expressed as the sum of two numbers of the form 6n-1 and 6n+1 - that’s trivial. But how do you then show that it’s always possible while stipulating that both numbers in your sums are primes?
Maybe you are on the verge of a simple, elegant, ingenious solution, but I don’t see it yet.
Indeed. Goldbach’s Conjecture claims that any even number can be written as the sum of two primes. But the converse of this—that any sum of two primes is an even number—is (assuming you exclude 2) trivially true.
The latter. Showing that even numbers are equivalent to 0, 2, …, 10 mod 12 is trivial and missing the point entirely. If you have a mathematical result, a general-interest message board is not the appropriate place for disseminating it. Write it up and post in on the arxiv, submit it to a journal, or at least post it on something like the stackexchange/overflow sites.
It’s an interesting approach but you made an error in the premise. While it’s true that all prime numbers can be expressed as either 6n+1 or 6n-1 it’s not true that all numbers that can be expressed as 6n+1 or 6n-1 are prime. As an easy counter example, there’s 25 which is not a prime but can be expressed as 6x4+1.
So while your list contains all the ways that you can have a sum of 6n+1 and/or 6n-1 and shows they’re all even, you can’t conclude from this that every even number is the sum of two primes.
Yes that I am aware not all numbers expressed in the form 6n+1 or 6n-1 are prime. That is why these equations alone are not enough to prove the conjecture. I am aware that the list will include some numbers that aren’t prime which would invalidate the proof.
However, I feel that you may have misunderstood what I am saying. So I will try and re-state it.
To try and explain it I will do a though experiment to try and illustrate what I am saying. Let us take all the potential differences that could exist between any two primes. This is an infinitely large list. Let us eliminate all the potential prime differences in the list that are either multiples of 6 or multiples of 4.
This will give a list of the prime differences: 2; 10; 14; 22 etc. In other words it will contain all the differences that can be expressed as 6(2n-1)-4 and those that can be expressed as 12n-2.
Now if you fiddle around with primes, always greater than three, that have these differences you can only create a multiple of 12. Therefore when these primes are added they will always create a multiple of 12.
You may think that this is obvious. 6n+1 + 6x-1 will always result in a multiple of 12. No that is an incorrect assumption. Prime differences that are multiples of 4 but not multiples of 6 when added will result in evens that can be written as 12n-6.
So it goes on and on. Now if you could prove that the equations were true for primes exclusively then it would prove the conjecture.
The equations I listed are not the differences between primes. I mentioned the differences between primes to show that I am not sucking this out of my thumb.
two prime numbers can only be expressed in a certain way depending on the difference between them. That is the first step.
The second leap is that these primes needn’t be consecutive such as 7and 11. It applies as well for that as 1069 and 5.
The third is that this only applies for primes that be expressed as either 6n+1 or 6n-1.
The fourth is a leap from step one. Now that I have 8 different ways that primes must be expressed depending on the difference between them how do I use that. Instead of using the difference between these two primes just add the two primes together. The result is the equations I mentioned in my OP.
The fifth leap is the conclusion. I am going to try and explain this again because this is causing the confusion I think. If I take the equations that are used to describe the primes for a given difference and add them all the results are unique.
Let me explain. Take the following differences between primes: 2; 10; 14; 22 etc. These differences are described in equations that I have mentioned in my previous post. Now depending on what the difference is there are specific equation to describe the larger and smaller prime. The equations aren’t as simple as simply 6n+1 or 6n-1 but for the case I am using as an example it makes no difference.
Now when these primes that have these differences are added they can only give result that is a multiple of 12. Forget Goldbach’s conjecture for a minute the maths I have used is 100% correct. I have double checked it and the maths is so simple that I could prove it on this forum. It is not hard. The maths tells me this.
For primes that have other differences between them the results are different. That is where the original equations I mentioned come from. I hope that helps.
Very true. This is not a proof for Goldbach’s conjecture for the reason you have mentioned. I haven’t proven it that I know. What I am saying is that depending on the even number only primes that have a difference that can be described in a certain way can be added create this even number.
So for example let us take 114. 114 can be expressed in the form 12n-6.
Therefore, ignoring 2 and 3 then only primes that have a difference between them that can be expressed as 6(2n-1)-2 or 6(2n-1)+2 can be added to create 114. So it goes on. It is rather a series of equations that defines mathematically the difference that must exist between for them to add to any given number. In the case of 114= 61+53. That is one example you will find it works for all primes greater than three. That is all I am saying.
Now The next thing I said was if in the far future some other person were to prove these statements could be the sum of two primes they will have proved Goldbach’s conjecture
You don’t have equations. You have expressions. Further, all you have really done is partition the even numbers into 6 separate equivalence classes for which you can’t generalize to “things that work for primes exclusively”. The partitioning is a bit too simplistic for that.
As I stated earlier, useful in a general sense? Sure. Useful in a specific sense? No. And certainly not really a new way of looking at integers. The notion of equivalence classes is pretty old now.
My recommendation: take a course on number theory and find a couple good texts on the subject. And definitely have some live chats with math people. The amount of interaction you can get through an internet message board is limited.
At the very least, this should give you an idea of what is known and not known about primes and provide some grounding on mathematical proof.
Many of the questions/ideas you’ve raised in this thread and in others are literally homework assignment problems in many number theory courses, so they really will help out a lot.
