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March 27, 2024

The twin prime conjecture

Filed under: Mathematics,the twin prime conjecture — Rob burchett @ 12:09 pm

As an entry into Concept Sharing 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.

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 two books packed tightly together, on a shelf.

Then the point of contact can be separated into two points, one for each item. Mathematically a single point is replaced by two distinct points, with a small gap, then this gap can be increased..

What if a point could be expressed as two items of no extent which were not points? Why does there only have to be one entity which has no extent?

Concept Sharing is a term used in education. But here I am giving it another definition.

As an entry into Concept Sharing 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.

What if a”point” could be expressed as two items of no extent which were not points? Since an item of no extent placed together with another similar item of no extent would still have no extent, yet there could be two items here. The two items would just be hidden as one. Why does there only have to be one entity which has no extent?

These two items could share the concept of a point (concept sharing). In order to do this we would have to take out the usual concept of a point and replace it with this new conception.

What does it mean to take out the concept of a point? It is not the same as removing a point from a given subset of points of the plane. We wish to open up a new possibility, so we have to reject the usual concept of a point and accept this new possibility. 

Yet to reject the concept of a point and replace it, it means we have to define a space where a point can be truly taken out of. A place of places. Another level to places. That is, a concept space. If you accept the notion of an entity with no extent you have to accept this new possibility as well.

To remove the concept of a point, we need to have a concept space, consisting of different nested levels of the concept. This must exist because I must have this further concept of a doubled item of no extent somewhere and I need to take out the concept of a single item of no extent to have it appear by itself (unconfused).

Further, all math concepts are point-like in that they are exact ideas which have no existence in physical reality. So number, set, group, function, etc. can all have concept spaces.

It seems like the two items should be the same. But not if they can be further defined in a new plane as separate.

With numbers, they represent a position, an amount or a label.

Think about position and consider a race where 2 runners are tied for 10th place. We can say the runners are tied for tenth place giving the number 10 to both runners. But suppose instead of a foot race, two points are together in a race along the number line. Then in 10th place two points could be together as has been seen. Then two number 10’s can be given one to each point. These numbers could go with the points as they are mapped into the new plane which has been mentioned.

A circle which is the 10th circle to be formed could contain the two e’s with the two number 10’s.

In the equation x^2+y^2=1 the two points (0,1) and (0,-1) can be mapped to the point (0,0).

This can be notated ….p(1)*p(2)*p(3)=p(1)*p(2)= p(0).

Where * is the idea of different points coming together in the usual plane.

When mapping to points there is a many to many map, a many to one map or a one to one map. Yet there is another possibility with e’s. There can be, for example, a two to two map. Where two separated e’s are in the new plane and combine to form two e’s at a single location. I say two points at that location now because they are defined in two different places of original places in the new plane. Therefore there are two here as in the case of the overlapping shadows.

Since it is possible to have two entities together and still be two entities, this other case must exist somewhere. The entities must be zero-dimensional but not be points.

Yet the overlapping shadows show us that there could be a constant number of entities. This is because the shadows have somewhere to be cast onto. Rather than a notion of no extent alone, the surface makes it possible to show a finite number of entities of no extent.

Then postulate a place of places where ‘e” the entity of no extent which is like a point in that it has no extent but unlike a point being different in the following way:

e(0) is not equal to e(1)*e(2) is not equal to e(1)*e(2)*e(3) where again * is the idea of movement but this time in the new plane.

Suppose I represent a location by (0,0). What if we can have a case of (0,0)(1) not equal to (0,0)(2)?. This is possible if (0,0)(1) and (0,0)(2) were somehow different. Since something with no extent can be “added” to something with no extent and the result is something with no extent, there could be two items here. (see the overlapping shadow diagram) We would have to somehow make the two “points” different and only two “points”.

So I say “added”, let us postulate another level of places. That is, an underlying plane where places of the usual plane may exist in other “placements” of places. Where a placement is not a place but a lower level of place. The same notion as place, yet let places be capable of shifting off into this new plane of placements. Then we no longer have a fixed plane of places, yet the placements could be fixed.

Since two items (“points”) of no extent could appear to be a single item, and we could conceivably fix this at 2 or three, or as many as we choose, this other plane also must exist.

The usual idea of points can be notated ….p(1)*p(2)*p(3)=p(1)*p(2)=p(0).This is the idea of two or more points coming together to form one point. This is all happening in the usual plane.

The other case can be notated e(0) not equal to e(1)*e(2) not equal to e(1)*e(2)*e(3). So e is not equal to p because it exists in placement space and has this other feature which is different from the way p behaves. In placement space we have 1 or 2 or 3 e’s together and also capable of being separated to different placements.

Then usually the idea of points can be notated pxp=p or pxpxp=p…etc. Where x is the idea of coming together and p is a point. But what if there were another entity of no extent, call it e such that exe=exe, e is not equal to p so that exe is not equal to p and also exe=is not equal to e as that would be the same as pxp=p. We can call these entity equations.

It seems like exe are two identical entities of no extent and it should result in e. But consider that to have exe=e, I have to take out p=pxp. This is not as simple as taking out a point out of a given subset of points of the plane as I have to be able to put something back in that is truly different.

This means I need a space of places coexisting with the original places, so that I can take out the place pxp=p and replace it with the new places, exe=exe. Then we give up the notion of a fixed plane of points, instead we can have three planes, coexisting with one another.

The most basic new plane is in a sense at a lower level than the usual plane. This is a plane of places of new places .Any e in the usual plane can move off in any direction into this new plane, leaving its partner behind. Most basically, the entire plane can move, as shown above.

That means exe are not 2e’s at the same place, as is usually thought of as place but two e’s at the same place in new places. A new level to place. Now we have more room. Since they are in this sense not in the same place, they don’t combine. Briefly we can write this exe=exe (sharing).

Take out the concept of place and put in this new concept of place. The only way it can be different is if the places don’t combine to form a single place but stay separate while being together. (sharing as in the overlapping teacup shadows)

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 different points of the shadows can take up the space of one point. This is analogous to two e’s sharing.

Then we can separate the two e’s, but the only way this can be different from the usual idea of separation in points, They are the same in that they share a place and a place of new and original places.

The place has been removed so we have an underlying dimension where places have other places of original places. Like a jigsaw puzzle of a landscape being taken apart. In this way two e’s are sharing a place and a place of new places.

Then this leads to a new extent, a line with two distances one being this new zero and the other being the usual concept of distance, extended.

This is a new dimension. Each e of the extent is different as any other e, yet they originally shared. This is just a new dimension in length. We can notate any two e’s as e(1,m) and e(1,n).

This extent may be considered as negative distance as we need to shrink it to get back to the new zero and then take this out and replace it with pxp=p to get back to the usual zero. Since for e(1,m) and e(1,n) the place is the same, any point that is bound to e(1,m) is also bound to e(1,n). Just not to both at the same time. We may have a closed loop of e’s which can move off and the shape could be altered if we have different distances associated with each e.

We can set a mathematical system with exe or choose three e’s so that we have exexe the number of e’s could be variable.

This must fit into our current structure of mathematics as I am not adding any new notion in, merely clarifying the concept of a point as having no extent, then adding in the necessary new entities. The notion of no extent is the same. We already have this notion of a point as being pxp=p, we have to extend this.

Additionally, there is also the case exr=exr where e and r are two different types of entities as well. This can be for future work.

So we have the idea that a point is an entity with no extent, and also another notion that it could be exe but how do these fit together?

It must be that we have replaced the usual idea of a point as being pxp=p with this new idea of a point as being exe=exe.. This means there is another level to space. Since I’ve taken out the usual notion of a point, I must have taken it out from somewhere. This is the space, places of new and original places. It can be modeled after the usual idea of extent, yet the distances are negative.

Then this also means I can separate exe=exe in the new space and move in a space between two of the same e, like so, the displacements from the 2e’s are shown.

Then we can have the idea of a multiple point or two tangent points.

With the tangent point we measure the diameters from the point of tangency outward. These can be separated as usual with the usual distance appearing between them. The exe=exe points can be separated as well, with the new space appearing between them.This is the space of places of new places. This is the negative space.

So we must have a plane or a space in which the ordinary places e or p, take on other places.

We could have a closed loop of these points all moving together as shown in the diagram above. As well, this loop could be knotted, if instead of a plane we consider a three dimensional space..

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 can all have sharings.

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:

A number is an amount, as in a counting number, or a position on a number line, or a label.

It is point-like in that it has no existence in physical reality, it is a mathematical object, not a physical object. Therefore we can make a correspondence between the idea of sharing in points and an idea of sharing in numbers.

So that means the concept of a number can be extended downwards so that we have a number of original numbers space and this number of sharing numbers after we take out the original number. So, for example, with the number 1; we have a number of numbers space, let the number of original numbers be 2, instead of 1. Take out the number 1, then we can have 1(1) and 1(2) sharing. This can be notated (1(1)(1(2)). We can show that there are really 2, 1’s there by calling it (1(1) like so:

(1)(x2) = (1<….(1) =(1(1).

The left 1 is really together with the right 1, yet I need to work with both of these so both of these are shown.

The twin prime conjecture:

There is a lower level of prime numbers using the concept sharing as it applies to numbers. 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 squares as shown, these are the primes and composites as seen with the extra added dimension. One is centered at (4(4). We connect the prime shared numbers.

There is a split between the natural numbers and the primes at (3(3).

We might also make copies of these prime’s associated with these squares. exe can be repeated indefinitely.  Yet there is a countable infinity of these possible. At the first of these new squares, there can be an indefinite amount of copies. Yet we can balance this infinity by stretching the copies out to the other infinity that is available to us.

The first time I encounter a new square type, there is no rule for how many copies exe=exe I have present. The only way I can have it exist is if there are an infinite countable number of copies present. Then I can move these copies into the other countable infinity I have present. The next such square is similar so that I do the same with this. Then this leaves a finite number of squares as we climb higher. This leaves a finite number of exe copies present at any junction. Then I can recover the usual numbers by replacing exe with pxp=p ie. (1(1) with 1.

The first small square I encounter, I spread out three primes and after that only composite numbers. The new definition of prime in the plane, is to have a prime sharing at every corner.

The second square is the new definition of a prime in the plane, with a prime shared number at every corner. There must be an infinite countable number of copies here as well. I have to be able to spread this out too, so I have an infinite number of twin primes, primes that differ by 2, and an infinite number that differ by 4, etc.

There is an infinite number of naturals (composites considered) and an infinite number of primes (larger squares considered)

The composites and primes can’t be anywhere else in the plane but on the midline. Since we need to recover the ordinary composites and 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 composite and prime numbers. Take away all the shared points except for one at each step. Then (1(1),(2(2),(3(3),(4,4),(5,5),(6,6),(7(7) can become 1,2,3,4,5,6,7…

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, as well as we see other gaps repeating.

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