# Interactive Online Tutoring Services

## August 4, 2023

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

As a basic introduction to a new geometry, consider two points existing together but not forming one point. This must be somehow possible as a point with no extent combined with another point of no extent will still have no extent, but there could be two points here. We have to introduce another level to geometry, since I have to take out the point that already exists in order to have this “sharing”. This means another level to space must exist. We also have to alter the idea of the number two as well. Do this we need to take the ideas of numbers and points and alter them (through “concept sharing”). I think these ideas are 17th century assessible. It may be that they were lost. See below for more details:

Online Tutoring Services Ontario Canada » a clearer and simpler explanation of knottedness using concept sharing (calctutor.ca)

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

Let’s start by going two steps back and considering an origin which may consist of an infinite number of these new points. Then bring a line segment out into two directions along a line. We restrict the space in order to do this, taking advantage of a line of places of places (where the places can move out into )and an area of places of places of places (where the places cannot go). It can have two integer lengths, yet a single length of lengths. Since length is not the same in the new geometry. It’s length of length might be one unit, but it’s lengths can be three or four units, for example. Then let one line segment, consisting of places and place of places be moved apart in the space of places of places of places. It has a fixed length of length so the only way we can extend it is by duplication and then movement. It is placed one after the other. Then this is the demonstration that a+b=c is possible for some cases of a, b and c.

Now we move to the next dimension and consider a square, the side length of which is again two possible integers. Let there be two sets of two squares making up the initial square. This is possible since we can have half points as well. If we use the same pattern as the case of one dimension this is the demonstration that a^2+b^2=c^2 is possible for some values of a, b and c. See the diagram below.

In three dimensions and higher this is not possible to do. In three dimensions I create six cubes instead of the required eight. Also see the sketch below. This shows the proof of Wile’s theorem (Fermat’s conjecture).