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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.

Overlapping shadows:

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).

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