### Goldbach’s conjecture

Abstract: Here I explain and use concept sharing. Then I take a look at Goldbach’s conjecture.

Overlapping shadows:

An interesting notion can come about from something observed in nature, the overlapping of two shadows. Consider a cup of tea placed on a table and lit with two lights from above, one from the left and another from the right, as illustrated below:

Three areas of shadows are formed. The one in the center is the overlapping of two shadows.

The concept sharing of a number:

Numbers are exact concepts. In the above case, we can think of them as the number of overlapping shadows at the center. They have exact boundaries and some way of showing we have two overlapping shadows there or three there, etc.

Then borrowing from the notion of overlapping shadows we should be able to hide numbers together and they would be “two hidden as one” as well. (concept sharing) if the mathematical objects represented by the numbers had the same boundaries, like the shadows at the center.

Other than the further darkness of the overlapping shadows, we cannot see or imagine that there are two separate shadows there. Similarly with two numbers hidden as one we can not see or imagine them together. Yet our logic tells us this can be so.

Then to this end let us create another number dimension, a dimension of number of numbers. Let the usual case be that the number of numbers is only 1. But now let us expand into the next dimension and allow the number of numbers to be 2.

So for example with the number 1, let us take away the original number 1 (since we have another underlying dimension, we can do this) and replace it with two new numbers 1(1) and1(2). These are together like the two shadows but do not form one number.

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects hidden as one. Or you can think of the two number 1’s as fitting into each other.

When we are numbering two mathematical objects which fit into each other, the objects are somehow different from each other and we give the two hidden objects two new numbers 1(1) and 1(2).

In the case of mathematical objects there is no external way of telling how many objects there are, previously it was assumed it was only one. We can state how many we wish at the onset thus fixing a certain mathematical system. Then we need the concept sharing of a number to indicate how many objects we wish to be there.

Goldbach’s conjecture:

We can think about new types of numbers (1(1)(1(2)), (2(1)(2(2)), (3(1)(3(2)),..These are higher numbers than the naturals, 2 being greater than 1. The notation is meant to show that the numbers are together in the number space, fitting into one another or hidden as one. I show a revelation of one number “peeking out” from “behind” another. Shown more clearly here: (a(a). For clarity and brevity I drop the (1),(2) designation from now on.

There are other numbers such as (1(2), (2(3), (1(3),… We will look at these later.

Consider the even shared numbers (2(2), (4(4), (6(6),…

We can break them down into a sum of two prime shared pairs such as (2(2)=(1(1)+(1(1) and also (4(4)=(1(3)+(3(1) if we define the other types of shared numbers with dissimilar basic numbers to fill in the rest of the lattice points. But (1(3) and (3(1) don’t seem to make sense since I can’t concept share two different numbers. Perhaps we can concept share at another level such as (1(1) shared with (3(3) then these could also be (1(3) and (3(1) by rearrangement.

Some numbers have more than one decomposition. Such as (6(6)=(3(3)+(3(3)=(1(5)+(5(1). In the modern understanding of primes 1 is no longer considered a prime.

But consider for a moment (4(4). I do not create it by writing one “4” and then another, since it is the combination of two numbers which is the number. So I have to work with the combination as much as I can. It can be created from the multiples or the sum of primitive numbers. These are the numbers which have to be formed by adding one basic number and then another in series.

Now think of the even shared numbers as being created from the primitives. Since I’m no longer creating the even shared numbers from the usual numbers on the number lines, these locations must come form somewhere else. These even shared numbers must go on indefinitely, as to create the numbers 2,4,6,… . Then the natural numbers are all at a lower level from these numbers. Also (2(2), (4(4), (6(6),…are derived from the primitives.

Then starting at (4(4), (4(4)=2*(2(2)=(1(3)+(3(1)=2*(1(3)=2*(3(1). So (2(2)=(1(3)=(3(1)!

Then also (6(6)=2*(3(3)=(1(5)+(5(1)=2*(1(5)=2*(5(1). So (3(3)=(1(5)=(5(1).

Also (8(8)=2*(4(4)=(5(3)+(3(5)=2*(3(5)

And so on. Each even shared number can be a multiple of another shared number but it must be built from a set of primitives.

Then if we look at one half of these binary decompositions we can see Goldbach’s conjecture is true. It was just a part of a deeper understanding of numbers.