### The twin prime conjecture

As an entry into Concept Sharing it’s best to start with a specific example. Then let’s start with the concept of a point. In math this is the notion of an entity with no extent, or in Cartesian geometry the notion of something with position only.

Consider a teacup placed on a table with two lights from above. One from the left and one from the right. See below:

Now as seen in the overlap, two points of the shadows can take up the space of one point.

We have the familiar idea of two items just touching or resting upon one another as we see in everyday life. For example a book resting on a table, or books packed tightly together, on a shelf.

But there is something else that is hinted at here. That is that points could be multiple. A point could be in combination with multiple copies of itself. There could be multiple copies of the one point present within the space of one point.

Look at the turtle diagram below. Points are tangent to one another, if we represent points with circles. This represents points which are together, which could be separated.

But we could also have multiplicity of the individual circles, representing multiplicity of points.

The trouble is that this must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying. We already have this notion of a point as being solitary, we have to extend this.

The solution is that since this other case must be possible, it must be possible to take out the point that is there and replace it with the possibly multiple point. I say possibly because we allow the case where the point could be single too.

This necessitates a “lower” level of places which contain original places, as points are thought of as places in Cartesian geometry.

So we must have a plane or a space in which the ordinary places of Cartesian geometry take on other places of places.

A picture of this would look like below:

This is a movement of one piece of a doubled origin.

Not only the origin but each identified point of the new space can act as its own centre, moving away from its copy.

We could have a closed loop of these points all moving together. As well, this loop could be knotted.

Then this is also the entry into Concept Sharing as math concepts such as number, set, group, ect. Can all be thought of as point-like. That is to say they are all ideas which could have multiple expressions.

They are all exact and have no physical reality, they are just ideas.

Since they can all be multiple, there must exist lower concept spaces.

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. The number then has a one-dimensionality to it as it is at least possible to separate it linearly.

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.

Since numbers are exact we can consider the objects to be like points. Since something with no extent placed onto something with no extent would still have no extent but there could be two objects here. Also we could think of a combined single object of no extent( at the start) created form two other objects of no extent.

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 concepts we wish to be present.

The twin prime conjecture:

There is a lower level of prime numbers using the concept sharing of a number. For examples: (1(1), (2(2), (3(3), (5(5),…I drop the (1), (2) notation for clarity and brevity.

Yet there is now a plane of numbers as shown below. With entries such as (3(5), (5(7), (5(11). These are partially shared prime numbers. We can make a squares as shown, with a gap of two on all sides, centered at (4(4). This connects the prime shared numbers.

We might also make copies of these prime’s associated with this square. Yet there is a countable infinity of these possible.

The shared prime pairs can’t be anywhere else in the plane but on the midline. Since we need to recover the ordinary prime’s, there can only be one copy of these. The rest of the copies are moved into the infinity that is available. Then in this way we are able to recover the prime numbers. Take away all the shared points except for one at each step. Then (1(1),(2(2),(3(3),(5,5),(7(7) can become 1,2,3,5,7…

Since I can make an indefinite amount of copies, the same gap must be possible infinitely as we go further up. Therefore the twin prime conjecture is seen. Also, we see how other gaps must be repeated too.