Category Archives: Last Fermat theorem

Three video’s to kill the time in case you are bored to the bone…

A couple of days ago I started on a new post, it is mostly about elliptic curves and we will go and see what exactly happens if you plug in one of those counter examples to the last theorem of Pierre de Fermat. There is all kinds of weird stuff going on if you plug such counter example in such a ‘Frey elliptic curve’. I hope next week it will be finished.

In this post I would like to show you three video’s so let’s start that: In the first video a relatively good introduction to the last theorem of Fermat is given. One of the important details of that long proof is the relation between elliptic curves and so called modular forms. And now I understand a bit better as why math professors go bezerk on taking such an elliptic curve modulo a prime number; the number of solutions is related to a coefficient of such an associated modular form. It boggles the mind because what do those other coefficients mean? As always just around the corner is a new ocean of math waiting to get explored.

Anyway, I think that I can define such modular forms on the 3D complex and circular numbers too so may be that is stuff for a bunch of future posts. On the other hand the academic community is never ever interested in my work whatsoever so may be I will skip that whole thing too. As always it is better to do what you want and not what you think other people would like to see. The more or less crazy result is shown in the picture below and after that you can see the first video.

Yet it might be this does not work on the 3D complex numbers…

Next video: At MIT they love to make a fundamental fool of themselves by claiming that their version of a nuclear fusion reactor will be the first that puts power on the electricity grid… Ok ok, after five or six years I have terminated the magnetic pages on the other website because it dawned on me that the university people just don’t want to read my work. I have explained many many times that it is just impossible that electrons are magnetic dipoles but as usual nothing happens.
Oops, wasn’t it some years ago that Lockheed Martin came bragging out they would make mobile nuclear fusion reactors and by now (the year 2021) there would be many made already? Of course I would never work properly because at Lockheed Martin they to refuse to check if the idea’s of electron spin are actually correct. If electrons are magnetic monopoles all fusion reactors based on magnetic confinement will never work. Just look at Lockheed Martin: So much bragging but after all those years just nothing to show. Empty headed arrogant idiots is whart they are.

And now MIT thinks it is their time to brag because they have mastered much stronger magnetic fields with their new high temperature superconducting magnets. Yes well you can be smart on details like super conducting magnets but if you year in year out refuse to take a look at electron spin and is that Pauli matrix nonsense really true in experiments? If you refuse that year in year out, you are nothing but a full blown arrogant overpaid idiot. And you truly deserve the future failure that will be there: A stronger magnetic field only makes the plasma more turbulent faster. And your fantasies of being the first to put electricity on the grid? At best you are a pathetic joke.

MIT & me, are we mutual jokes to each other?
Just like ITER and the Wendelstein 7X this will not work!

It is very difficult to make a working nuclear fusion reactor on earth if you just don’t want to study the magnetic properties of electrons while you try to contain the plasma with magnetic fields. Oh the physics imbeciles and idiots think they understand plasma? They even do not understand why the solar corona is so hot and if year in year out I say that magnetic fields accelerate particles with a net magnetic charge, the idiots and imbeciles just neglect it because they are idiots and imbeciles.

The third video is about a truly Hercules task: Making a realistic model of the sun so that can run in computer simulations… If humanity is still around 10 thousand years from now may be they have figured it out but the sun is such a complicated thing it just cannot be understood in a couple of decades. There is so much about the sun that is hard to understand. For example a number of years ago using the idea that electrons are magnetic monopoles, it thought that rotating plasma like in some tornado kind of structure is all you need to get extremely strong magnetic fields. But I never ever wrote down only one word in that direction. Anyway about a full year later I learned about the rotational differential for the sun: at the equator it spins much faster as it does on the poles. And that would definitely give rise to a lot of those tornade like structurs that must be below the sun spots.
Of course nothing happens because of ‘university people’ and at present day I do not give a shit any longer. I am 100% through with idiots and imbeciles like that. For me it only counts that I know, that I have figured out something and trying to communicate that to a bunch of overpaid highly absorbed in their giant ego’s idiots and imbeciles is a thing I just stopped doing. If it is MIT, ITER or Max Planck idiots and imbeciles, why should I care?

Ok, that was it for this post. If you are not related to a university or academia thanks for your attention. And to the university shitholes: please go fuck yourselves somewhere we don’t have to watch it.

On a beautiful identity related to the new little theorem of Pierre de Fermat.

Just a few minutes ago I thought that may be the next title to this new post would also be funny: WTF! Doing math research with the help of a Google search suggestion??? And yes, if you think about that it is rather weird but it is true. How come I have fallen so deep?
Well a couple of weeks back I did a Google internet search by typing in one of my counter examples to the last theorem of Pierre de Fermat. In the normal html search results nothing popped up but when I looked in the pictures to my surprise every time there was only one picture of my website. And indeed that picture contained the search phrase I had just typed in… So at present date & year the Google search algorithm can translate a phrase like 5^n + 7^n = 12^n mod 35 to the actual content of a picture on this website. By all standards this is amazing. Here are the two search phrases I did put in and if you do that yourself you can find back in the pictures from Google search the actual picture on this website that the search phrase is about. Here are the two pictures explaining the stuff a little bit:

This is amazing!

When I started this website in 2015 in the second post I explained how this website was set up: The math will be mostly in the pictures I create so it is a pity that internet search engines cannot read that. Well now almost six years later as a matter of routine the Google search engine can indeed a bit of the math content as found in my pictures.
Here is the second search string or search phrase:

To be honest I was amazed at the results. So later I decided to check what would pop up on Google if I would search for my new little theorem of Pierre de Fermat. So my search phrase was p^q = p mod pq and yes Google can find that content in my pictures too.
But the Google search engine also popped up an alternative search phrase also containing the mod pq stuff. And for a few seconds of time I started to panic like ‘Oh oh have the math professional professors already found the new little theorem of Fermat and am I only an idiot doing weird stuff that is known to science for centuries????‘ But very soon I calmed down, ok they might have proven that identity that Google search suggests, but they are still overpaid weirdo’s never ever able to connect this particular identity to the new little theorem of Fermat. Compare it to the Cauchy-Riemann equation that rule differentiation in the Complex plane; despite having those 2D equations the math professors just cannot bring those same ideas to 3D space. And not for just a few years or so, no they cannot do that century in century out. Even if there is a dedicated website out like this website doing a lot of 3D complex number theory, the overpaid & incompetent math professors keep on being silent year in year out and likely also century in and century out.

But let’s not get emotional about how stupid this all is, after the indentity found is very beautiful and all in all very much like my new little theorem of Pierre de Fermat. And may be it is a bit unwise from me to view the collective of math professors as ‘utterly stupid’ After all they have some kind of proof for it and most of those proofs use the CRT or the Chinese remainder theorem. And that made me pause for a moment, the remainder theorem is very old so how the hell did China industrial spionage in other parts of the world in those long lost centuries? Did they find the CRT without stealing intellectual property? Wow!

But let’s get serious, this post is five pictures long and all about that search suggestion as done by Google. Before we start I once more want to show you that the new little theorem of Fermat is indeed an improvement. For those who already know that, just skip it if you want.
The old theorem of Fermat says that a^p = a mod p, here p is a prime number and p should not be a factor of a. So a and p must be relatively prime to each other. But if p is a small number, it does not work that well. Take for example a = 105 and p = 2. Well 105^2 mod 2 = 1 but everybody already knows that the square of an odd number is odd.

The new little theorem of Fermat says a^p = a mod ap and because ap is always greater than a, this also returns a when p is small. For example now we have 105^2 mod 210 = 105. So I think this is a small improvement.

Ok, let us go to the five pictures:

Yes it is the end of this post but I would like to remark that if you look at the banner of this website it says ‘Math as you have never seen it before’. Of course that is not a goal an sich but a natural by product of the things I like to study. My math all hang together in ways that are often not obvious. For example the new little theorem of Fermat is not a stand alone result, it emerges directly from when I studied the exponential orbits during all those counter examples to the last theorem of Fermat. In return all those counter examples started with the first ones I found in my own 3D complex numbers. So this all hangs together while understanding this new little Fermat theorem can also be understood using just plain old math without any 3D complex numbers or so.
The professional math professors never found this new little Fermat theorem because they do not want to walk the path from 3D complex numbers to where we are now. They only want the 2D complex plane and ok may be the cute quaternions and that’s it. Of course that is not forbidden, but if in 3.5 centuries of time you even cannot find the new Fermat little theorem, might it be time to scratch the head a little bit and do some rethinking?

Of course not my dear reader, the math professors will never change.

See you in the next post & thanks for your attention.

The new little theorem of Fermat: Checking it out with Pari.

This week the first blue berries were ripe so it is that time of the year again: Beer with blue berry juice! Lovely; it is a very smart way to get a bit of vitamins…;)
It is not much of a secret that every now and then I have some tiny forms of criticism against the professional math people. It really is minor and tiny; calling somebody an inverted asshole is often even a compliment compared to what they actually deserve. This day it is different, I came across a computer program named Pari and it is actually very good. I think or estimate this program started before we had computers with a so called GUI (=graphical user interface, you know the click stuff with a mouse). So you have to type in commando’s and press enter.
There are two versions of the program; a 32 and a 64 bit version. On my old computer the 32 version does not work properly (it does not work at all but that is more my computer to blame and not the program). The 64 bit version runs perfectly so that’s fine.
Pari seems to be used a lot by number theorists, I do not know much about number theory so that is why I never heard of the program. Pari is from France so if you speak that language will you can even install the french version of it all. For myself speaking I am glad I now finally have some program that does not limit me to the say 12 digits of a Casio hand held calculator. Not that you need so many digits very often but it is good to have that capability.

One of the other reasons for posting this is that when you do a Google search for ‘The new little theorem of Fermat’ my relevant posts from earlier this year just do not pop up on page one, two or three of the search results. On the one hand that is logical: a lot of people and websites with large traffic volumes have been writing about the little theorem of Pierre de Fermat. I would like to let it pop up a bit higher in the search results so you readers must all massively do a search every day on Google for the new little theorem of Fermat…;)
I remember that when I started this website, if I searched for 3D complex numbers you always had that stuff known as Alan’s pages (or may be Allen’s pages) popping up above me. Just always. That guy had only one page on 3D complex numbers and everything you could do wrong, Alan did it wrong. So for me it was kind of frustrating; why does that idiot pop up above me year in year out? It was only later that I realized a lot of people click on that kind of fake science because those people likely did everything wrong themselves. Likely they are professional math people who are trying to expand the complex plane to 3D space, that is something that does not work at all. If you do not base it on an imaginary unit, say j, with the property j^3 = -1, all your work will be in vain. Anyway after some time I finally popped up above that stupid Alan page and since that time it has always been higher. The lesson is you must not hurry; math is a long term thing and with this new little theorem of Pierre de Fermat it will take some time before it will be valued properly.

The Pari program is very good at doing the modulo thing, if you want to calculate a mod b in Pari you must enter the command Mod(a, b).

This post contains four written pictures about the new little theorem of Fermat, I left the proof out because I already wrote two proofs, after that just a few screen shots from the program Pari. And of course a few links for the downloading stuff although the Pari program is easy to find on the internet. Let me start with four pictures in the size 550×825 pixels:

This website is a WordPress website and as such it is horribly bad at handling larger image files. So I hope and pray that the screen shots that are about 850 pixels wide will display properly. Here we go, this is the opening screen of Pari and you see it is just like a DOS window:

As you see it does not display properly; it looks like it is 550 pixels wide so likely that is how I ended with those math pictures all 550 pixels wide…

A detail not to be missed is that you can get moral support and all you have to do is type the command ?17. That is very handy, so the next time you feel frustrated you do not need to go out and slaughter and kill at least 20 children and babies in order to get rid of your frustration. No, all you have to do is type ?17. It is amazing! No more baby killing, just type ?17.
Ok, next screenshot that will not display at 850 pixels wide:

Beside moral support, this is how you can get help.

In the above screen shot you see why I like this Pari computer program: You can fill in calculations like 8269^9973 for the a in Mod(a, b) and 8269*9973 for the b in Mod(a, b). Remark that 8269 to the power 9973 is a very huge number, yet Pari gives the correct answer in a split of a second.

Let’s look at the last screen shot:

In the above screen shot you see a numerical validation of the new little theorem of Pierre de Fermat. Of course we do not need such validations because I already gave two proofs for the new little theorem. But it is nice to observe that a standard computer can handle giant numbers like 8269^9973 with easy. So my compliments go to the French math community for this perfect computer program!

Here is a link to the Pari homepage: https://pari.math.u-bordeaux.fr/
Do not skip the documentation, it’s all not bad.

Ok, let me try to find that button known as ‘Publish website’ and see you around in a new post.

Oversight of all counter examples to the last theorem of Pierre de Fermat, Part 3.

It is late at night, my computer clock says it is 1.01 on a Sunday night. But I am all alone so why not post this update? This post does not have much mathematical depth, it is all very easy to understand if you know what split complex numbers are.
In the language of this website, the split complex numbers are the 2D circular numbers, In the past I named a particular set of numbers complex or circular. I did choose for circular because the matrix representations of circular numbers are the so called circulant matrices. It is always better to give mathematical stuff some kind of functional name so people can make sense of what the stuff is about. For me no silly names like ‘3D Venema positive numbers’ or ‘3D Venema complex numbers’. In math the objects should have names that describe them, the name of a person should not be hanged on such an object. For example the Cayley-Hamilton theorem is a total stupid name, the names of the humans who wrote it out are not relevant at all. Further reading on circulant matrices: Circulant matrix.
I also have a wiki on split complex numbers for you, but like all common sources they have the conjugate completely wrong. Professional math professors always think that taking a conjugate is just replacing a + by a – but that is just too simplistic. That’s one of the many reasons they never found 3D complex numbers for themselves, if you do that conjugate thing in the silly way all your 3D complex math does not amount to much…
Link: Split-complex number.

This is the last part on this oversight of counter examples to the last theorem of Pierre de Fermat and it contains only the two dimensional split complex numbers. When I wrote the previous post I realized that I had completely forgetten about the 2D split numbers. And indeed the math results as found in this post are not very deep, it’s importance lies in the fact that the counter examples now are unbounded. All counter examples based on modular arithmetic are always bounded, periodic to be precise, so professional math professors could use that as a reason to declare that all a bunch of nonsense because the real integers are unbounded. And my other counter examples that are unbounded are only on 3D complex & circular number spaces and the 4D complex numbers so that will be neglected and talked into insignificance because ‘That is not serious math’ or whatever kind of nonsense those shitholes come up with.

All in all despite the lack of mathematical depth I am very satisfied with this very short update. The 2D split numbers have a history of say 170 years so all those smart math assholes can think a bit about why they never formulated such simple counter examples to the last theorem of Fermat… May be the simplicity of the math results posted is a good thing in the long run: compare it to just the natural numbers or the counting numbers. That is a set of numbers that is very simple too, but they contain prime numbers and all of a sudden you can ask thousands and thousands of complicated and difficult questions about natural numbers. So I am not ashamed at all by the lack of math depth in this post, I only point to the fact that over the course of 170 years all those professional math professors never found counter examples on that space.

This post is just 3 pictures long although I had to enlarge the lastest one a little bit. The first two pictures are 550×825 pixels and the last one is 550×975 pixels. Here we go:

That was it for this post, one of the details as why this post is significant is the use of those projector numbers. You will find that nowhere on the entire internet just like the use of 3D complex numbers is totally zero. Let’s leave it with that, likely the next post is about magnetism and guess what? The physics professors still think there is no need at all to give experimental proof to their idea of the electron having two magnetic poles. So it are not only the math professors that are the overpaid idiots in this little world of monkeys that think they are the masters of the planet.

Oversight of all counter examples to the last theorem of Pierre de Fermat, Part 2.

Post number 191 already so it will be relatively easy to make it to post number 200 this year. If you think about it, the last 190 posts together form a nice bunch of mathematics.
In this post we will pick on where we left it in the last post; we start with the three dimensional complex and circular numbers. In the introduction I explain how the stuff with a pair of divisors of zero works and from there it is plain sailing so to say. When back in Jan of this year I constructed the first counter example to the last theorem of Pierre de Fermat I considered it a bit ‘non math’ because it was so easy. And when one or two days later I made the first counter example using modular arithmetic I was really hesitant to post it because it was all so utterly simple…
But now half a year later it has dawned on me that all those professional math professors live up to their reputation of being overpaid under performers because in a half year of time I could find not one counter example on our beloved internet. And when these people write down some calculations that could serve as a counter example, they never say so and use it only for other purposes like proving the little theorem of Fermat. It has to be remarked however that in the past three centuries of time, when people tried to find counter examples, they likely started with the usual integers from the real line and as such tried to find counter examples. Of course that failed and this is not because they are stupid or so. It is the lack of number spaces they understand or know about that prevented them in finding counter examples to the last theorem of Pierre de Fermat.
If you do not know anything about 3D complex or circular numbers, you are not a stupid person if you cannot find counter examples to the last theorem. But you are definitely very very stupid if you do not want to study 3D complex numbers, if you refuse that it proves you have limited mathematical insights and as such likely all your other math works will be limited in long term value too.
While writing this post all of a sudden I realized I skipped at least one space where counter examples are to be found: It is on the space of so called split complex numbers. I did not invent that space, that was done by the math professors. The split complex numbers are a 2D structure just like the complex plane but instead of i^2 = -1, on the split complex plane the multiplication is ruled by i^2 = 1. Likely I will write a small post about the split complex number space. (Of course in terms of the language of this website, the 2D split complex numbers are the 2D circular numbers.)

This post is 8 pictures long, I kept on to number them according to the previous post so we start at picture number 11. They are all in the size of 825×550 pixels. I hope it is worth of your time. Here we go:

In this post I used only ‘my own spaces’ like 3D complex and circular numbers and the 4D complex numbers. As such it will be 100% sure the math professionals will 100% not react on it. Even after 30 years these incompetents are not able to judge if there is any mathematical value in spaces like that. Why do we fork out so much tax payer money to those weirdo’s? After all it is a whole lot of tax payer money for a return of almost nothing. Ok ok a lot of math professors also give lectures in math to other studies like physics so not all tax payer money is 100% wasted but all in all the math professors are a bunch of non-performers.

I think I will write a small post about the 2D split complex numbers because that is a space discovered by the math pro’s. So for them we will have as counter examples to the last theorem of Pierre de Fermat all that modulo calculus together with the future post on the split complex numbers. Not that this will give a reaction from the math pro’s but it will make clear you just cannot blame me for the non reactive nature of the incompetents; the blame should go to those who deserve it… Or not?

May be the next post is about magnetism and only after that I will post the split complex number details. We’ll see, anyway if you made it untill here thanks for your attention and I hope you learned a bit from the counter examples to the last theorem of Pierre de Fermat.

Oversight of all counter examples (until now) to the last theorem of Fermat.

Once more I estimated my writer skills too positive; I thought it takes only 7 to 8 pictures to pen down all counter examples to the last theorem of Pierre de Fermat. But I had to stop because this post already is 10 pictures long. I organized the stuff around so called ‘levels’ so we have Level zero counter examples (the most basic ones) to level four counter examples (the most complicated ones) to the last theorem of Fermat.
We are now almost half a year further down the timeline since I looked at that video of Andrew Wiles and while looking at that video I already constructed the first counter example to the last theorem. And now we are about half a year further I have arrived at the conclusion that inside the math community there are no accepted counter examples known or discussed or whatever what they do with it. So I am not expecting my counter examples will find the place they more or less should have, acting as an adult is never within reach of the math professors.
Likely if you (as a math student or so) show these counter examples to one of your math professors, it might be it gets instantly rejected because ‘this is modulo calculus’ and has ‘nothing to do with the last theorem’. But in a pdf from Gerhard Frey (the guy from the Frey elleptic curve) I came across the next hilarious screen shot:

This is hilarious; Modulo arithmetic is allowed?????

Of course people like Andrew Wiles or Gerhard Frey will never ever react; they are perfumed princes and those high nobility people do not mingle with the plebs, farmers and peasants.
Does the absence of counter examples on the internet indicate that the math hot shots also have an absence of counter examples in their holy brain tissue? It looks like it, of course it is very well possible plenty of people found those counter examples in the past but they got only a fresh dose of what I experience for say 30 years: You get one 100% neglected all of the time. After all if you are an outsider like me, the collective of math professionals acts very much like a secretive incest club. Those kind of clubs are not very open to communication with the outside world. Not that math professors are pedophiles but the behavior is keeping things inside is very much the same.
An interesting question is if perfumed prince Andrew Wiles knows these counter examples. My guesstimate is no. And why is that? After the pictures of the main post I will show you that video from the Abel prize lecture and there you can see Andrew formulating the last theorem of Pierre de Fermat. He does it more or less in the following manner:

For integers x, y and z with xyz not equal to 0, it is impossible that
x^n + y^n = z^n. This for integers n > 2.

If Andrew would have knowledge in his brain about these counter examples likely he would have formulated it like this:

For positive non-zero integers x, y and z, it is impossible that
x^n + y^n = z^n. This for integers n > 2.

All counter examples I found in the last five months have the property that xyz = 0 because say the x and the y form a pair of so called divisors of zero. (The level zero counter examples do not have this property, but I did not find the level zero examples myself…) Now it could be that Andrew is as evil as he is smart and formulated it with the xyz not zero way in order to subtlelly exclude my wild math adventures. But in reality it is much more likely that Andrew is just another perfumed Princeton professor. So my estimate is and stays that Andrew is not aware of the rather simple counter examples to the last theorem of Pierre de Fermat as found below…

Ok, this post is 10 pictures long and it only contains the stuff found that is based on modular arithmetic. In my view it is all rather basic and as such should explain itself. The main basis is that ‘divisor of zero’ stuff where two non zero numbers multiply to zero. For example if we take the real integers modulo 35, in that case if we multiply 5*7 we get 0 because 35 = 0 inside the ring of integers modulo 35. It’s all so basic that it makes you wonder once more: why is this not inside the math classes on universities? But for that answer you must ask the perfumed professors and the perfumed professors never answer because they are much more busy of getting a fresh stack of perfume. That’s life…
Have fun reading it

For me the only thing I learned while writing this post is that you can use that simple rule for writing 30^n mod 210 in stuff mod 7. It is the same kind of calculation as in the modified little theorem of Fermat and this is also a detail I could not find on the internet.
Oh yes now I am in a laughing mood: Can we prove the perfumed professors are indeed perfumed overpaid people? Yes we can: They will also not react of the modified or improved little theorem of Fermat. We will need a few years of time for that, but why should they react?

Ok, I promised you the video of a Princeton perfumed prince named Andrew Wiles, here it is and please pay attention to the way he formulates the present day formulation of the last theorem of Pierre de Fermat:

At only one minute into the video we observe that likely Andrew has no clue whatsoever about the above counter examples to the last theorem of Pierre de Fermat:

Why this way of formulating the problem?

End of this post. Thanks for your long lasting interest in these details on the last theorem of Pierre de Fermat.

Two things and a proof that the 4D complex rationales form a field.

I finished the proof that was originally meant to be an appendix to the previous post. And I have two more or less small things I want to share with you so lets get started with the first thing:

Thing 1: Tibees comes up with a very cute program of graphing 3D surfaces. It’s name is surfer, the software is very simple to use and it has the giant benefit of making graphs from implicit equations like
f(x, y, z) = 0. For example if you want the unit sphere in 3D space you must do x^2 + y^2 + z^2 – 1 = 0. Now for this website I always used an internet applet that uses ray tracing and by doing so over the years such graphs always look the same. But this surfer program has cute output too and it has the benefit you don’t need to be online. Here is how such a graph looks, it is the determinant in the space of 3D complex numbers, to be precise it shows the numbers with a determinant of 1:

By the way, the surface of this graph is a multiplicative group on it’s own in 3D space. I never do much group stuff but if you want it, here you have it. And for no reason at all I used GIMP to make one of those cubes from the above graph. It serves no reason beside looking cute:

The Tibees female had a video out last week where she discusses a lot of such surfaces in three dimensional space using that surfer software. And she is a pleasant thing to look at, it is not you are looking at all those extravert males drowning in self-importance only lamentating shallow thoughts. The problem posed in the video is an iteresting one, I don’t have a clue how to solve it. Title of the video: The Shape No One Thought was Possible. It is a funny title because if you start thinking about all the things that math professors thought were not possible you can wonder if there is enough paper in the entire universe to write that all out..
Link to the Surfer program in case you want to download & install it:
https://www.imaginary.org/program/surfer.

So far for thing 1.

Thing 2: The last weeks it is more and more dawning on me that all those centuries those idiots (the math professors) did not find counter examples to the last theorem of Fermat. Nor was there any improvement on the little theorem of Fermat. Only Euler did some stuff on the little theorem with his totient function, but for the rest it is not much…
Well since Jan of this year I found many counter example to the last theorem of Fermat and in my view I made a serious improvement on the little theorem of Fermat.
So is the improvement serious or not?
Here is a picture that shows the change:

So it’s modulo ap instead of modulo p.

On a wiki with a lot of proofs for the little theorem of Fermat they start with a so called ‘simplification’. The simplification says that you must pick the number a between 0 and p. So if you have an odd prime, say a = 113, does the little theorem only make sense for exponents above 113?
And can’t we say anything about let’s say the square 113^2?

With the new version of the little theorem we don’t have such problems any longer. Here is a screen shot from the start of that wiki, the upper part shows you the improvement:

Here is a link to that wiki that is interesting anyway.

If you follow that link you can also scroll down to the bottom of the wiki where you can find the notes they used. It is an impressive list of names like Dirichlet, André Weil, Hardy & Wright and so on and so on. All I want to remark is that non of them found counter examples to the last theorem nor did they improve on the little theorem of Fermat. Now I don’t want to be negative on Dirichlet because without his kernel I could never have crafted my modified Dirichlet kernel that is more or less the biggest math result I ever found. But the rest of these people it is just another batch of overpaid non performers. It’s just an opinion so you don’t have to agree with it, but why do so many people get boatloads of money while they contribute not that much?

End of thing 2.

Now we are finally ready to post the main dish in this post: the proof that the subset of four dimensional rational numbers form a field. Math professors always think it is ‘very important’ if something is a field while in my life I was never impressed that much by it. And now I am thinking about it a few weeks more, the less impressed I get by this new field of four dimensional complex numbers.
Inside the theory of higher dimensional complex numbers the concept of ‘imitators of i‘ is important: these are higher dimensional numbers that if you square them they have at least some of the properties of the number i from the complex plane. They rotate everything by 90 degrees or even better they actually square to minus one.
Well one of the imitators of i in the space of 4D complex number is dependent of the square root of 2. As such it is not a 4D rational complex number. That detail alone severely downsizes my enthousiasm.
But anyway, the next pictures are also a repeat of old important knowledge like the eigenvalue functions. Instead of always trying to get the eigenvalues from some 4×4 matrix, with the eigenvalue functions with two fingers in your nose you can pump out the eigenvalues you need fast. This post is six pictures long each size 550×825 pixels.
Here we go:

Yes that is the end of this post that like always grew longer than expected. If you haven’t fallen asleep by now, thanks for your attention and don’t forget to hunt the math professors until they are all dead! Well may be that is not a good idea, but never forget they are too stupid to improve on the little Fermat theorem and of course we will hear nothing from that line of the profession…

Inverses for the field of 4D complex rationales.

This year starting in January I found more and more counter examples to the last theorem of Fermat. As a by product when we looked at the stuff on the 4D complex numbers, we found that if we restrict ourselves to the 4D rationales, they were always invertible. And as such they form a field, this is a surprising result because the official knowledge is that the only possible 4D number system are the quaternions from Hamilton. So how this relates to those stupid theorems of Hurwitz and Fröbenius about higher dimensional complex numbers is something I haven’t studied yet. But that Hurwitz thing is based on some quadratic form so likely he missed this new field of 4D complex rationales because the 4D complex numbers are ruled by a 4 dimensional thing namely the fourth power of the first imaginary unit equals minus one: l^4 = -1.
Compare that to the complex plane that has all of it’s properties related to that defining equation i^2 = -1.

And because we now have a 4D field I thought like let’s repeat how you find the inverse of a 4D complex rational number. And also prove that we have a field as basic a proof can be. But while writing this post I had to abandon the second thing otherwise this post would grow too long. Of course in the past I have crafted a post for finding the non-invertible 4D complex numbers but in that post I never remarked that rational 4D complex numbers are always invertible. To be honest in the past it has never dawned on me that it was a field, for me this is not extremely important but for the professional math people it is.

When back in Jan of this year I found the first counter example to the last theorem of Fermat I was a bit hesitant to post it because it was so easy to find for me. But now four months further down the time line I only found two examples where other people use some form of my idea’s around those counter examples and both persons have no clue whatsoever that they are looking at a counter example to the last theorem of Fermat. But in a pdf from Gerhard Frey (that is the Frey from the Frey elliptic curve that plays an important role in the proof to the last theorem of Fermat by Andrew Wiles) it was stated as:
(X + Y)^p = X^p + Y^p modulo p.
That’s all those professionals have, it is of a devastating minimal content but at least it is something that you could classify as a rudimentary counter example to the last theorem of Fermat. It only works when your exponent if precisely that prime number p and it lacks the mathematical beauty that for example we have in expressions like:
12^n = 5^n + 7^n modulo 35.

Anyway this post contains nothing new but there is some value in repeating how to find inverses of higher dimensional complex numbers. All you need is a ton of linear algebra and for that let me finish this intro on a positive note: Without the professional math professors crafting linear algebra in the past, at present day for me it would be much harder to make progress in higher dimensional complex numbers. And it is amazing: Why is linear algebra relatively good while in higher dimensional number systems we only look at a rather weird collection of idea’s?
This post is made up of seven pictures each of size 550×800 pixels.

Stupid typo: Z = 1 + l +… so the real part must be one.

Ok ok this post is not loaded with all kinds of deep math results. But if you have a properly functioning brain you will have plenty of paths to explore. And the professional math professors? Well those overpaid weirdo’s will keep on neglecting the good side of math and that is important too: That behavior validates they are overpaid weirdo’s…

For example the new and improved little theorem of Fermat: The overpaid weirdo’s will neglect it year in year out.
That’s the way it is, here is once more a manifestation of the new and improved little theorem of Fermat:

Let’s leave it with that. Thanks for your attention.

A second proof of the new little theorem of Monsieur Pierre de Fermat.

This is not a totally new proof, basically it is already part of the proof in the previous post. But I was able to write it down a bit more compact. It seems that you can skip large parts of the previous proof and still the result is standing. The stuff you can skip is not unimportant or so; it is where I compared those additive and multiplicative orbits and you really need that in order to understand a bit more of what is happening in rings like the integers modulo 35 (or any other composite number of course).
Often I name the ordinary intergers ‘real integers’. I hope that is not confusing, with a real integer I mean whole numbers as they are found on the real line. I do that do make clear the difference with the Gaussian integers as they are found in the diverse complex spaces.

Right now I am already about four months busy with this stuff that all started in January with counter examples to the last theorem of Fermat. All in all I never expected to be able to make an improvement on the little theorem of Fermat. But sometimes the old little theorem gives answers that are indeed correct but still is not that satisfactory. In the old little theorem you take some real integer a, you pick a prime number p that has no common factors with a and you know that in that case:
a^p mod p = a. But if the prime number p is relatively small, I mean a is larger, you don’t get back a but a mod p.

This post is short, only three pictures in the usual format of 550×775 pixels but I added two appendices so all in all there are five pictures in this post. In the second appendix I show you that if we square 125, the old little theorem gives back a 1 while my new version of the little theorem nicely gives back 125. Of course there is always much discussion possible of something is ‘better’ or not. It is only in the sense that also with small prime numbers you get back your a I mean it is ‘better’. One thing is clear: it is definitely more beautiful. My favorite formulation stays the formulation with two prime numbers like in:

p^q = p mod pq &
q^p = q mod pq.

The symmetry in the pair of equations above is, in my opinion, more beautiful compared to the old version of the little theorem of Fermat. At last I want to remark that I have a thing in common with Monsieur Fermat: math is a hobby for me. Now the old little theorem was improved upon by the professional professor Euler and with a little smile on my face I can say: Hey Euler did you miss the above pair of cute equations?

Yes he did, just like a guy named Einstein never had a fucking clue about electron spin. But likely that is a story for another day, in the meantime we have five pictures with the second proof of the new little theorem. Have fun reading it and never forget: If it is math, sometimes you need a few more days to figure it all out!

In a wiki upon proofs of the little Fermat theorem you can find that one of the authors did indeed find a counter example to the last theorem of Fermat. You can find it under ´Multinomial proofs´in the next link:
Proofs of Fermat´s little theorem. Link used:
https://en.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem

Ok, that was it for this post. Thanks for your attention.

Proof of the conjecture on Fermat’s little theorem.

The proof is finished and in the end it went rather different from what I expected before the writing down of this new proof. I hope the main ideas are easy to understand. I formulated the proof with a concrete example; so not two general prime number p and q. But I took p = 5 and q = 7 and as such we are calculating in the ring of integers modulo 35.
One of the key ideas is that we have so called additive orbits, for example the additive orbit of 5 is the set {0, 5, 10, 15, 20, 25, 30}. The additive orbit are just the multiples of 5, it is handy to view the above set as multiples of five and as such: {0, 5, 2*5, 3*5, 4*5, 6*5}.
On the other hand we have exponential orbits, the exponential orbit of 5 are the powers of 5 like in the next sequence: 5, 5^2, 5^3, 5^4
An important observation is that any power of five is also a multiple of five; that means the exponential orbit is inside the additive orbit. For example 5^3 is on the exponential orbit, 5^3 = 125 = 20 modulo 35 = 4* 5.
So the number 5^3 from the exponential oribt corresponds to 4*5 on the additive orbit. The goal of the proof is to show that the period of 5 in her exponential oribt is 6 and the period of the exponential orbit of 7 is 4.
That will ensure our new little theorems of Fermat:
p^q = 5^7 = 5 modulo 35 &
q^p = 7^5 = 7 modulo 35.
Remark that powers of 5 can never be a multiple of 35 simply because any power of 5 does not contain a prime factor 7, so not all of the numbers on the additive orbit are allowed.
Another key idea is that if we reduce stuff modulo 35, this is the same as reducing stuff modulo 7 on the multiples of 5. Let me explain: Take the number 50, inside the mod 35 ring this is 15. But 50 = 10* 5 = (10 – 7)*5. I was able to pull that modulo 35 stuff on a ring back to modulo 7 stuff on a field…

Another thing I want to remark is that I formulated these new little theorems of Fermat mostly in prime numbers. That makes them more symmetric like the beautiful pair of equations above, but it can be a bit more general like I showed you in the last post using the number 210 that is made up of four different prime factors. And even that is not needed; prime factors can be double or triple it does not matter. As long as the exponent is a prime number my freshly crafted proof will sail you through all the troubles there are.

In a parallel development I found a perfect math professor. It’s a female and she has an amazing career record: At age three already a Fields medal while before she was nine years of age already the third Abel prize for lifetime achievement math… It is surely amazing…;)

Unlike other math professors, she is thinking it all through… Amazing!

But serious, this post is not that long. Only five pictures and like I said above it is not a ‘most general’ kind of proof but it uses a fixed pair of prime numbers. I think it is better this way because if I would formulate all the stuff in a general setting the only people who could understand such a writing are the ones who already figured the stuff out for themselves… Ok, all pictures are in the usual 550×775 pixelf format so here we go:

We are almost at the end: Let me give you one more example as why only in the exponent you need a prime number. We take the number 8 and raise it to the power 5 and do the reducing modulo thing by 40.
Doing so gives that: 8^5 mod 40 = 32768 mod 40 = 32768 – 819*40 = 8.

Ok, that is what I had to say for this post.