Interactive Online Tutoring Services

March 27, 2024

The twin prime conjecture

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

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. 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 square 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.

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.

March 13, 2024

Showing the Culprit knot is the unknot

Filed under: knots,Mathematics,unknotting the Culprit knot — Rob burchett @ 2:26 pm

Starting with the diagram of the Culprit knot, I am trying to find a way of showing that I can form the unknot without increasing the crossing number.

I do so by placing the knot in placement space. The the locations are free to move in a space of locations of locations. We keep the locations connected in the knot as they were originally connected.

Specifically, they can move around in a loop along the path of the knot.

Once we decompose two crossings into a joining (alpha-beta’s), we can double up the knot diagram again. One diagram is still and the other I can move the locations around again along the path of the new knot diagram. Then I can decompose again, etc. This means I can move one alpha-beta past another.

I then decompose completely and look for another way to put the knot back together. This new diagram is obtainable from the original diagram by the usual Reidmeister moves.

Contact us today to get started