# Interactive Online Tutoring Services

## January 26, 2024

### Goldbach’s conjecture

Filed under: goldbach's conjecture,Mathematics — Rob burchett @ 4:26 pm

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

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.

## December 29, 2023

### General Math Concept Sharing

Filed under: general math concept sharing,Mathematics — Rob burchett @ 10:00 am

General Math Concept Sharing

For Concept Sharing we needed to start with sharing numbers. The idea is to form sharing concepts. For this we need these new numbers, to be specific so that we can start with certain mathematical systems.

In mathematics, concepts are mental constructions. They are ideas like shadows or ghosts without boundaries. They can be thought of like points.

We can concept share a point since two items of no extent will still have no extent, but there could be two items here.

So similarly numbers, as we have seen, points, sets, groups, ect. Can all concept share.

In order to do this we must remove the concept which is present initially and replace it with 2 or more sharing concepts. Since it is possible to share concepts there must exist a more underlying concept space.

The two sharing concepts must be different in some way which we can specify based on the nature of the concept itself.

Then an infinite level concept space can form as I can continue into the next level of the concept and so on. One may utilize as many levels as is necessary.

Additionally there can be a finite number, an infinite countable number of an infinite uncountable number of sharing concepts, as this is the understanding of numbers.

## November 5, 2023

### Introduction to Concept Sharing and knots

Filed under: introduction,knots,Mathematics — Rob burchett @ 1:16 pm

Concept Sharing and a New View of Knots

Abstract: Here I introduce concept sharing. In uncovering extended space, I show a new way of understanding knots.

An interesting notion can come about from something observed in nature, the overlapping of 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 shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.

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. Two or more numbers can be hidden as one since natural numbers represent exact positions.  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. This must already exist since it should be possible to put two or more numbers together at a beginning. 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 two numbers together can be notated ((1’(1)((1’(2)). 1’(1) is “peeking out” from behind 1’(2). Shown by the use of a half parentheses. Seen more clearly here: (a(a).

Keep in mind that these numbers are different. They do not represent two obviously separate objects, but represent two mathematical objects, also concept sharing, hidden as one.

The objects are concept sharing as well and are somehow different from each other. 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.

A New Plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to place two together using two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)

An object of no extent created together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system.

The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension.

Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so each point of interest of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a subset of sharing points co-existing with a plane of places of places.

One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. In this way these two can be different. Then all of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.

## September 5, 2023

### Work for introduction

Filed under: introduction,knots,Mathematics — Rob burchett @ 12:27 pm

### The knot equivalency moves

Filed under: knot equivalency moves,knots,Mathematics — Rob burchett @ 12:24 pm

### The Riemann hypothesis

Filed under: Mathematics,the riemann hypothesis — Rob burchett @ 12:22 pm

It seems to me there may be an easier way to express the zeta function: Z(s)=1/1^s+1/2^s+1/3^s…..using the ideas of concept sharing as it applies to a new geometry.

Concept Sharing and a new view of the Riemann hypothesis

Abstract: Here I introduce concept sharing. In uncovering extended space, I develop new ways of understanding the Riemann hypothesis.

An interesting notion can come about from something observed in nature, the overlapping of 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 shadows at the center. They have exact boundaries and some way of showing we have two there or three there, ect.

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.

The objects are somehow different from each other. 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

A new plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to place two together using two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)

An object of no extent placed together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system.

The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension.

Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so each point of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a whole plane of doubled points co-existing with a plane of places of places.

One of the new points can be fixed, while the other one is capable of “shifting” away in this new dimension of place. The places of places line can be defined in a space of places of places of places. Then a place of place, with a place, can move in the upper half-plane. Make a small jump back to the places of places line. We can remove the places in this small gap then remove other places as motion continues as the line underneath is revealed.

Then some of the sharings in the new plane can become new origins-one point being fixed while the other point is capable of shifting away.

The Riemann Hypothesis:

One may imagine a type of gird with the first square being 1, the next being 1/2^2 the next being 1/3^2… if we use s=2 as an example. See pictures in the notes below. The higher numbers of s can be seen by increasing the dimension. Yet there is always a plane possible with any dimension equal to or higher than 2.

Since with concept sharing geometry there comes a place of places, in which places can vary, we may vary the distance as we choose to always make the zeta function defined. The zeta function can be continued into the extended geometry. Then there is no longer a need for analytic continuation. I can always make the grid into a 1 by 1.

So we can create a grid specific to the Zeta function defined in placement space.

Then we have that there are two types of number involved. A real part and an imaginary part.

I think this can be seen more primitively as a numbers which lead to a square with a positive area and numbers which lead to a square with negative area ie. the negative distance is -i. These can be sharing space.

We can concept share two different numbers in the following way: (-1(-1)*(-i(-i) where * is a concept sharing of a concept sharing= ((-1(-i))((-1(-i)). But -1 and -i have to be different. Let -i be the negative distance and -1 be the other, real distance. Then let this be how the square comes about. We have to expand the zero-dimensionality of the concept sharing. Let -i and -l be numbers at the next level of numbers. That is they are no longer point-like but line like. We can start with a point consisting of an infinite uncountable number of sharing parts and expand it outwards into a line.

As in the next to last note the square is formed with two zeros. One at each corner. Then after real 1 is matched with imaginary 4 and real 2 is matched with imaginary 3,…

Then it comes about that the topology of the real part with a definite center is clearly 1/2 of the whole, if I fold over the square to match the zeros. The real parts add using coincidence. The complex part comes about by adding together linearly all the complex pieces that add by concept sharing. This shows the solutions of the extended Zeta function to have a real part of 1/2 and a variable imaginary part when seen in the complex plane.

The last image shows how there are trivial zeros at -2,-4,-6… and how the zeta function could equal -1/12 when s=-1. We are adding an infinite series to get a finite sum. This comes about as we have a addition of positive and negative area. This works for the plane as we can have i and i^2=-1.

## August 4, 2023

### A clearer and simpler demonstration of Fermat’s last theorem (Wile’s theorem)

Concept Sharing and Wile’s Theorem

Abstract: Here I introduce concept sharing. In uncovering extended space, I show a demonstration of Wile’s Theorem.

An interesting notion can come about from something observed in nature, the overlapping of 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 shadows at the center. They have exact boundaries and some way of showing we have two there or three there, etc.

They are ideas or concepts so that they are further like the shadows in that there is no real substance to them.

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.

The objects are somehow different from each other. 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.

A new plane:

Points are also exact concepts. In the Euclidean plane they are places, with the notion of no extent, in the plane. We should be able to place two together using two new numbers 0’(1) and 0’(2) identifying that we have two points. (0 is indicating an origin)

An object of no extent placed together with another object of no extent, would still have no extent- but there could be two objects here, under another mathematical system.

The two points 0’(1) and 0’(2) can be different by first uncovering a new place dimension, a place of places. This space can be thought of as akin to a jigsaw puzzle being taken apart over an underlying space.. This must already exist because there must be some way to have two points exist together and still be two points.

In a similar way as we uncovered the new number dimension (the number of numbers) we can uncover the new place dimension.

Take the original point out (we can do this since we have a new underlying dimension of place, a place of places) and replace it with the two new points. This can be done for the whole plane of points.

That is, there is nothing special about the origin, so each point of the usual plane can be removed and we can replace it with a “sharing” of two points. So that we have a whole plane of doubled points co-existing with a plane of places of places.

Wile’s Theorem:

Now the statement of Wile’s theorem is that the sum of two squares can equal a square, but the sum of two cubes or any higher power cannot equal a single cube, a fourth power or higher (more widely known as Fermat’s Last Theorem).

It seems then that it should be possible to demonstrate this with geometry. One of these new geometries mentioned above is a possible way of demonstrating this.

Let’s start by considering a line of places of places defined in a plane of places of places of places and a line segment which can consist of two or more superimposed lengths of places (two or more lengths).

At the start we can only have two types of points, fixed or mobile. Let the places of places be the fixed points, then since we can move off into two directions we must have 2 line segments with one 1 point each one line moving left and one line moving to the right. It can have two integer lengths (or multiple lengths), yet a single length of lengths which can vary. Since length is not the same in the new geometry.

It’s length of lengths might be one unit, but its lengths can be two, three or four units, for example. It’s lengths can only be multiples of the length of lengths and the length of lengths can vary.

Then let one line segment, consisting of two different sets of places and place of places be decomposed (simplified) in the space of places of places along the line of places of places. It has a length of length.

We can only move out in two directions along this line. It is seen that it is only possible to have two different places of places at the beginning. The places of places are mobile, and they can only move out left or right. So we double the mobile points and weight each one point, since I want to form the sum of two lines.

Suppose we map these two lengths of lengths co-linearly, inside the original by shrinking each line. Then this is the demonstration that a+b=c is at least possible for some cases of a, b and c. a, b and c being some lengths. Since the sum of two lengths of lengths is also a length of length as well.

Then this at least makes it possible that a+b could equal c. b may be too small or too big and not equal c, but there may be a case when a+b could equal c. Now the intention is to move up in dimension.

Now we can move to the next dimension by rotating the line of places of places out of the line and into a plane. When perpendicular we have a square, the side length of which is again two possible integers. Let there be a set of two squares making up the initial square, I can only have fixed or mobile points. Then since I can move off into four different compass directions n,e,s,w. This one mobile square must be made of two squares and must be rated at ½ points each.

Since I must move the copied squares out into an area of places of places it must be following the parallel lines which are places of places. I can move out four possible squares.

This indicates that I am moving the sum of two squares out to become four squares, which means the points of the squares are weighted ½ each. Then map these squares and move them all into the original square. See the diagrams below.

If we use the same pattern as in the case of one dimension this is the demonstration that a^2+b^2=c^2 is at least possible for some values of a, b and c , since the summed squares can add to a square in some cases. I can start with one square and add area around that square, which adds up to a square to try and form a final square.

In three dimensions and higher this is not possible to do. In three dimensions I create six cubes instead of the required eight. Each of the six cubes can be weighted ⅓ but we cannot form an added cube, since I need 8 cubes to do this. See the sketch below.

In a fourth dimension I would also not have the required number of hypercubes and so on. This shows a geometric proof of Wile’s theorem (Fermat’s conjecture).

## July 31, 2023

### Unknotting the Culprit knot-page 1

Filed under: knots,Mathematics,unknotting the Culprit knot — Rob burchett @ 2:12 pm

### Unknotting the Culprit knot-page 2

Filed under: knots,Mathematics,unknotting the Culprit knot — Rob burchett @ 2:11 pm

Thanks to Lou Kauffman for sending me a picture of the Culprit knot.

### Unknotting the Culprit knot-page 3

Filed under: knots,Mathematics,unknotting the Culprit knot — Rob burchett @ 2:09 pm
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