An old unsolved problem regarding the exponential function f(x) = e^x.

This is a problem I found about thirty years ago and I was never ever able to solve it. The problem as I formulate it is about finding a so called ‘composition root’ to the exponential function. Just keep it simple, say the composition ‘square root’. If we denote that as r(x) what I mean is that this function if composed with itself gives the good old exponential function: r(r(x) = f(x) = e^x.
There are many interesting aspects to this problem. For example take a piece of paper and a pencil and draw the graph of the exponential function and the identity function. It is now very easy for every point on your graph of the exponential function to find the graph of the double composition f(f(x) = e^(e^x)). But, as far as I know, you cannot go back and given the function f(x) find it’s composition root r(x).
It is very well possible that this problem is solved in the theory of dynamical systems. If memory serves we once had a lesson in when a family of functions could be interpolated but that was 30 years back and what I want is explicit expressions and formula’s and not only a vague existencial proof without a way to find an explicit answer.

Back in time before the logarithm was invented, the people of those long lost centuries had a similar problem understanding what exponential behaviour was. And you can go a long way in understanding exponential behaviour but say for yourself; without knowledge of the logarithm that kind of knowledge is far from being optimized.

In this post I only talk about the composition square root but of course any n-th root should be possible and as such giving rise to the idea that you can iterate or compose the exponential function also a real number amount of times. I have to admit I also have no proofs for the solution to this all being unique, but you should be able to differentiate all stuff found and it should still be coherent so my gut feeling says the solution is unique. My guess is there is only one ‘composition square root’ r(x) that is as smooth as f(x) itself…

This post is only two pictures long so here we go:

And it is also the end of this post. Give it a thought and if you are able to make some inroads on this that would be great. But all in all I think we do not have the math tools to crack this old old problem.

See you around in the next post.

Five highlights of the year 2021.

Despite my slowly detoriating health the last year was a remarkable fruitfull year when it comes to new stuff. So I selected five highlights and of course that is always a difficult thing. Two of the highlights are about magnetism and the other three are just math. Once more: The fact that I include two magnetic highlights does not mean I am trying to reach out to the physics community in any meaningful way. If these idiots and imbeciles keep on thinking that electrons have two magnetic poles, be my guest. There is plenty of space under the sun for completely conflicting insights: Idiots and imbciles thinking that electrons have two magnetic poles and more moderate down to earth people that simply remark: for such a bold claim you need some kind of experimental evidence that is convincing.
But 2021 was a very good year when it came to math; I found plenty of counter examples to the so called last theorem of Pierre de Fermat. I was able to make a small improvement on the so called little theorem of Fermat. A very important detail is that I was able to make those counter examples to the last theorem so simple that a lot of non math people can also understand it. That is important because if you craft your writings to stuff only math professors can understand, you will find yourself back in a world of silence. Whatever you do there is never any kind of response. These math professors were not capable of finding three or four dimensional complex numbers, they stay silent year in year out so I have nothing to do with them. In the year 2021 I classified the physics professors to be the same: Avoid these shitholes at all costs!

After having said that, this post has eight pictures of math text and it has the strange feature that I am constantly placing links of posts I wrote in the last year. So lets go:

Below you find the link to the 01 Jan 2021 post:
Once more: Zero reaction from the overpaid idiots & imbeciles.
This is the tau for the three dimensional circular numbers! Not for 3D complex numbers.
Next link contains the proof of the improved little theorem as posted on 20 April:
Here is that perfect animated gif once more:

I think that if you show the above animated gif to a physics professor and ask for an explanation, likely this person will say: “Oh you see the electrons aligning with the applied external magnetic field, this all is well understood and there is nothing new under the sun here”.
Of course that kind of ‘explanation’ is another bag of bs, after all the same people explain the results of the Stern-Gerlach experiment via the detail that every electron has a 50% probability that it will align with the applied external magnetic field (and of course 50% that it will anti-align). In my view that is not what we see here. As always in the last five+ years an explanation that electrons are magnetic monopoles with only one of the two possible magnetic charges is far more logical.

This year in the summer I wrote an oversight of all counter examples to the last theorem of Pierre de Fermat I had found until then. It became so long that in the end I had three posts on that oversight alone. I wrote it in such a way that is starts as easy as possible and going on it gets more and more complicated with the counter example from the space of four dimensional complex numbers as the last example. So I finished it and then I realized that I had forgotten the space of so called split complex numbers. In the language of this website the split complex numbers are two dimensional circular numbers. It is just like the complex plane with two dimensional numbers of the form z = x + iy, only now the square of the imaginary unit is +1 instead of i^2 = -1 as on the complex plane. So I made an appendix of that detail, I consider this detail important because it more or less demonstrates what I am doing in the 3D and 4D complex number spaces. So let me put in one more picture that is the appendix of the long post regarding the oversight of all counter examples found.

I hope this brings some clarity to the minds of math people.

All that is left is place a link to that very long oversight:

Ok, so far for what I consider the most significant highlights of the previous year. And oops, since I am a very chaotic person before I forget it: Have a happy 2022! It is time to say goodbye so think well and work well my dear reader.

Two videos so bad they are actually funny & a PERFECT gif found.

If you start commenting on bad videos you will have a busy hobby for the rest of your life. But there are also reasons to take a look at these videos, for example the math video is horrible but the path of calculation shown is rather beautiful. The other video is about magnetism and when I viewed it for the first time it was really late at night and only after a good night sleep I realized how horribly bad that video was.
But it was the magnetism video that made me look up the average size of the so called magnetic domains and that was when I found that PERFECT gif. So I cannot say it was all a waste of time, that perfect gif is made with something that is named a Kerr microscope and with such a device you can make magnetic domains visible.
Years ago, if memory serves it was Feb 2017, I was studying so called ‘racetrack memory’ that was under development by IBM. That IBM project failed because they kept on hanging to electrons being ‘tiny magnets’ with two magnetic poles, because that is likely not true all their work failed. Anyway they came up with the fact that you cannot move magnetic domains with magnetic fields and I totally freaked out. Late at night I realized that within my broader development of understanding magnetism at the electron level, the IBM findings were logical if magnetic domains in say Iron or so, always have a surplus of either north pole monopole electrons or south pole monopole electrons. Domain walls separate the two kinds of magnetic domains. Itis a pity that about five years back I never heard of those Kerr microscopes.
Again I want to highlight that I do not want to convince anybody that electrons are the long sought magnetic monopoles. I have done that for six or seven years and it was only in this year 2021 that I arrived at the conclusion that physics professors are just as stupid as the average math professor. It is a pile of garbage so it is not much of a miracle that six or seven years of trying to apply logic did not work at all. So from this year on going into the future the physics professors have the same status as the math professors: A pile of rotten garbage that you must avoid at all times at all costs needed.

After having said that, this post is five pictures long where I comment on the two horrible videos. Below that I will post the two videos so you can see for yourself (or may be you want to see them first). And at the end you can see that perfect gif where magnetic domains change in size due to the application of an external magnetic field. Also back in 2017 I more or less figured out how magnetic domains will change if you approach a piece of iron with a permanent magnet. What you see in the gif is more or less precisely that: Some domains grow while domains next to that shrink.

Ok, here we go:

Now we can go to the first video, the math one:

I found the magnetics video by doing an internet seach on ‘The Stern-Gerlach experiment for iron’. It is disappointing that almost no significant results are there. Some of stuff out of the 2030-ties of the last century but that was all behind pay walls. Very high in the rankings came the next video that uses iron filings to mimic or imitate the Stern-Gerlach experiment. The video guy should have used magnets on only one side, if that resulted into attraction & repulsion of the iron filings he would have gotten a standing ovation from me. Without any insult; the way he executed this experiment is a true disaster only showing he does not understand why the SG experiment is so important.
And by the way: If my idea of electrons being magnetic monopoles is in fact correct, you do not have to use inhomogeneous magnetic fields. Everything will do; even the most constant magnetic field in space and in time will do. But again after so many years of talking to deaf ears from stupid physics people, I have lost my desire to convince anybody any longer..

With magnets on two sides; of course it will spread out! This is stupid!!!!!

Ok, I have never hung any animated gif into this WordPress website so let’s check it out if it works properly:

As you see: Some domains grow while adjecent ones shrink.

I found this animated gif in a wiki: Magnetic domain.
That was it for this post. Thanks for your attention.

Video: What happens to Fermat’s last theorem modulo a prime number p?

This year in January I started finding more and more counter examples to the last theorem of Pierre de Fermat. I started in the space of 3D complex numbers using the so called divisors of zero you can find there. Rather soon after that I found those very easy to understand counter examples using modular arithmetic modulo N where N was the product of two prime numbers. Along the way I found out there is also a cute improvement to make to the so called little theorem of Fermat, that was one of the many cute results of this year. Simple example:
Take 107^2, the old little theorem of Fermat says you must take it modulo 2 and you get the number 1. With the improved little theorem you do not take it mod 2 but mod 2*107 or mod 214. And voila: 107^2 mod 214 = 107.
So the problem of the little theorem of Fermat with small exponents is more or less solved in this way. It was a cute result for sure…:)

Yet since January I have been wondering if all counter examples must use some form of divisors of zero. If so there would be a small possibility to strongly improve on the result from Andrew Wiles. The video below shows a counter example using only integers modulo a prime number p. Since those number spaces modulo p do not have divisors of zero, actually these spaces are fields, I can now say that it is not needed to have divisors of zero. So that is why I decided to include this video in this post: it saves me a lot of time and funny enough this result goes back to at least the year 1917.

Lately there are much more math video’s on the usual video channels and a lot of them have these beautiful animations. A significant part of those new video’s can be traced back to the work of the guy that runs the 3Blue1Brown channel who more or less started to craft those beautiful animations. Somewhere this summer or may be spring he called for the public to make their own math video’s and even I thought about it a short time. But I have no video editing software, I have zero experience in using such software, I have no microphone and the only video camere I have is my photo camera that I never use to make video. So I decided not to try and make a video about 3D complex numbers, it was just too much work.
Here is how the small picture from the video looks on Youtube, it looks cute but likely it takes a lot of time to make such animations:

That graph sure looks cute!

Ok, back in the year 1917 a guy named I. Schur managed to find a counter example using the integers modulo a prime number. He himself in that old article cites another person so it is a bit vague how this all panned out. Anyway I do not have to be ashamed as not to find this result modulo p; it is based on the colorings of graphs like you see above. I do not know much about that so don’t blame me for missing these results. Besidew that, another cute result of this year was that I found out that my own 4D complex numbers form a field if you restrict them to the rational numbers. That simply disproves stupid but accepted theorems like the Hurwitz theorem that says that the only higher dimensional complex numbers are the quaternions and total garbage like the octonions. That whole fucking stupid theorem from Hurwitz is just not true and of course the mathematical community says nothing.
Anyway, the video is just 10 minutes of your time and my post is only two pictures long so have fun with it!

And now we go for the video:

The paper that was unpicked is written in German but don’t panic: It is not in that gothic font they used for a long time. You can find it here:
So if you can read the German language you can try to unpick it for yourself… Ok, end of this post. Thanks for your attention.

What happens if an electron beam hits a magnetic field? Five easy calculations around magnetism.

It is just past midnight on a Saturday so why not start this new post? Two weeks ago all of a sudden I felt like doing a magnetics post once more and to be honest that is a tiny miscalculation because this is also post number 200. This should actually be about higher dimensional complex numbers or so but it is what it is so we are doing magnetics.
This year I stopped working on the magnetic pages on the other website, after more than five years and zero response I decided to classify the physics professors the same as the math professors: Just another bunch of incompetents that you better avoid being around. That policy stays in tact so although this post is very long you should not view this as an attempt to change the views physics professors have on magnetism in particular the idea that electrons must be magnetic monopoles because all other interpretations are not logical. To be honest I feel a bit more free now, now I can say you are a total idiot or a complete moron or a natural born imbecile instead of every time trying to come up with more reasons as why electrons cannot be magnetic dipoles.
This post contains five easy calculations around magnetism, to be a bit more precise: there are three forces that can act on an electron (Coulomb, Lorentz and say the Stern-Gerlach force) and the five easy calculations cover that. Ok, gravity also works on electrons but that is out of this story.
We will look at the speed and the acceleration of electrons and we will try to estimate how much of a gradient a magnetic field must have to explain the observed experimental results. I found a gradient of about 450 million Tesla/meter and that all in my living room… I cannot find a fault in it and of course a gradient so large is not realistic so I reject the way electron potential energy related to magnetism is incorperated into the standard model of particles.
Last week I found a perfect photo that can serve as a model of how physics professors are behaving in my view (a bunch of overpaid incompetents). The ‘blah blah blah’ can serve as my five easy calculations; for the idiots and imbeciles that the physics professors are, these calculations will be ‘blah blah blah’ material. Come on, next year 2022 it will be one 100 years since the Stern-Gerlach experiment was performed. If a collection of overpaid idiots have it wrong for one 100 years, the most logical thing is that they will keep it wrong for at least another 100 years.

Excluded the picture above, this post is 14 pictures long. And even then I skipped a lot of stuff like when talking about the solar wind I just assume all particles go with the same speed of say 750 km/sec. Of course that is an oversimpification; there are many different particles in the solar wind that all behave rather different.
I hope you don’t loose oversight and therefore I would like to explain how I arrived at that crazy magnetic gradient of 450 million Tesla/meter.
Here we go:

1) I estimate how fast electrons go in an old color television where I assume these electron have a kinetic energy of 40.000 eV. This is about 120 thousand km/sec or about one third the speed of light.

2) If you place a stack of those strong neodymium magnets at the television screen you see a black spot where no electrons land. I estimate the sideway acceleration the electrons make and that is a crazy huge number: about 4.5 times 10^15 m/sec.
That might sound very huge but it is just simple Newtonian mechanics: If the electron has a sideway acceleration this big for about 2 nanoseconds, the sideway displacement is about 1 cm. That is in line with what the photo of the television screen says..

3) I calculate the force on the electron for this sideway acceleration and plug this into the expression the professional physics professors use for the force related to inhomogeneous magnetic fields and voila: there is your gradient of only 450 million Tesla/meter.

All in all the five calculations are say advanced high school level / first year university level. Given the fact that physics professors keep on thinking that electrons are tiny magnets with a north and a south pole we can safely conclude that even a science like physics is much more some kind of social construct and not a hard ball science in itself.

Enough of the introductory talk, here are the 14 pictures:

Source of the in-picture screen shots:

Ok, that was it for this post. Please remark that the polar aurora’s do exist for real while the blah blah blah of physics professors explaining this makes not much sense. For example they do not explain why the electrons go so fast that they ionize the air and most of all: These idiots are not aware they are missing something here.

At last I want to remark that for me it feels rather refreshing to talk about physics professors as idiots and imbeciles. That is much better as always being polite and respectful. Why should overpaid idiots and imbeciles also face a lot of completely misplaced respect?
See you in the next post & thanks for your attention because this post was a long reading for you.

De Padé approximations; why are they so good?

Already a few years back I wanted to write a post on the so called de Padé apprximations because they are so good at taking the logarithm of a matrix. For me the access to an internet application that calculated those logs from matrix representations was a very helpful thing to speed things up. It would have taken me much much longer to find the first exponential circle on the 3D complex numbers if I could not use such applets.
But in the year 2018 pure evil struck the internet: the last applets or websites having them disappeared to never come back. Ok by that time I had perfected my method of simply using matrix diagonalization for finding such logs of matrices. You can still find it easily if you do an internet search on ‘Calculation of the 7D number tau‘.
Yet in the beginning I only had such applets as found on the internet and I soon found out that using the so called de Padé approximation always gave much better results compared to say a Taylor approximation.
It is not very hard to understand how to perform such a de Padé approximation. Much harder to understand is how de Padé found them, after all it looks like a strike of genius if this works. The genius part is of course found in the stuff you can simply neglect in such approximations, at first it baffles the mind and later you just accept it that you are more stupid as de Padé was…;)
Anyway this week I stumbled upon a cute video and as such I decided to write a small post upon this de Padé stuff. (On the shelf are still a possible new way of making an antenna based on the 3D exponential circles and some updates on magnetism.)
So let us first take a look at the video, here we go:

As you see the basic idea is pretty simple: you use those two polynomials to ‘approximate’ that Taylor series and as a bonus you have a much better approximation of the original function. All in all this is amazing and it makes you wonder if there are methods out there that are even better compared to this de Padé approximation.

Now you can choose beforehand what degree polynomial you use in the nominator and denominator. There are plenty of situations where this brings a big benefit like in the video they point out the divergence problems of say the sine function that is bounded between +1 and -1 on the real axis. The Taylor approximations always go completely beserk outside some interval where they fit quite well. With the appropiate choice of the degrees of the polynomials in the de Padé approximation you can avoid this kind of stuff.
In my view the maker of the video should also have pointed out that a de Padé approximation can have it’s own troubles when you divide by zero. And when the original function never has such a pole at that point, the de Padé approximation also goes very bad. These de Padé approximations
are indeed much better compared to the average Taylor approximation but they are not from heaven. You still have to use your own brain and may be that is a good thing.

De Padé still rules in the year 2021, likely it was a good idea to start with.

In this post I did not cover the matrices and why a de Padé approximation of the logarithm of a matrix is good or bad. If you want to find exponential circles and curves for yourself, use those applets mostly on imaginary units who’s matrix representation has a determinant of +1. In case you want to find your very first exponential circle, solve the next problem:

Ok, it is late at night so let me hit that button ‘publish website’ and see you around.

A infinite list of counter examples to the last theorem of Pierre de Fermat.

This is a very simple post, all it contains is a list of the most simple counter examples to the last theorem of Fermat. You might wonder, if it is that simple, why take the trouble and post it? Well sometimes it is important to stress the obvious and say for yourself: there is not much reaction from the math professionals so although it is utterly simple it could be that even that is once more too difficult…

These most simple counter examples are all based on two (different) prime numbers and how they behave under exponentiation while taking that modulo the product of those two prime numbers. The most simple example is given by 2^n + 3^n = 5^n mod 6. For n you can only take natural numbers because inside the ring of integers modulo 6 the two prime numbers 2 & 3 do not have an inverse and as such 2^(-1) and the likes simply do not exist. Remark such counter example via some exponential orbit are always periodic, so each and every counter example to the last theorem of Fermat only adds a finite number of ‘actual’ counter examples. In the above example the period is 2 because for n = 3 we already have 2^3 + 3^3 = 5^3 = 5 mod 6.
I organized the list in a simple manner: We like at pairs of prime numbers say (p, q) and the q is always the plafond or maximum prime allowed.
For example the plafond q = 13 yields the following pairs: (2, 13), (3, 13),
(5, 13), (7, 13) and (11, 13).
The reason for doing so is that it is now very obvious it is a one dimensional list of counter examples to the last theorem of Fermat. You can now, if you wanted it, make a one to one correspondence between the set of natural numbers and the list of counter examples.

Another reason for writing this post was that I wanted to experiment a bit with other backgrounds in the pictures used. In the list of counter examples I used two of those beautiful photo’s from the Nikon small world contest. If you have never seen them, look it up on the internet. Althugh those Nikon photo’s are very beautiful I do not think they form a good background to the math as published in the futute so I stick to my old backgrounds I just guess. I always make my math pictures with a very old graphics program that only runs on windows XP. My graphics program is so old that when I save something as a jpg format picture, the program always informs me how long it will take to send it over 28kB telephone modem… Yes, not even a 56 kB modem but a 28 kind of modem. That shows more or less how old my graphics program is but making those kind of backgrounds with say a modern version of GIMP is extremely hard and very laborsome. I like GIMP too but that old program still has features that GIMP is bad at delivering.

Another problem is the picture size under use in a WordPress website. When I started this website about six years ago I could not use any longer my favorite size of 550×1100 pixels. It did not display properly. Anyway only now six years later I dived into that problem again and it seems that WordPress makes 3 pictures of every picture you upload:
1) A thumbnail,
2) A display picture that is not ‘too large’ and
3) Your original upload visible after clicking on the display picture.

There are all kinds of issues with that; if one of my original uploads contains an error and I correct it, that never shows up on the display pictures. So I repair a fault but it does not show up. Only when I rename my faulty picture the correction of the fault is there. But that is frustrating on a lot of levels because now the natural naming of my math pictures gets distorted for no reason at all. For example if a picture has the file name 11Aug2021=Frey_elliptic_curves_are_stupid05.jpg and I have to repair a fault that is in it, I have to make the name different from the other pictures and that is weird. You can only delete your original upload and not the extra pictures that WordPress generates. In that sense it is welcome to the Hotel California where you can checkout any time you like but you can never leave.
If my expectations are correct, the first picture will display properly while the second one is likely too small but if you click on it you will see the correct version of stuff:

This displays properly…
You see, it cannot handle a picture that is 1440 pixels high…

Why my version of WordPress does not show the larger picture is unknown to me. Likely I must change some settings but I do not have a clue. Luckily it is still readable so may be this is a good time to say goodbye and until the next post and so.

What happens to a Frey elliptic curve if we take it modulo 35?

Of course we have to take a Frey elliptic curve based on one of the counter examples to the last theorem of Pierre de Fermat for this ‘take it mod 35’ to be meaningful.
Welcome by the way!

A few months back I encountered this problem for the first time and the determinant of all of those Frey elliptic curves is always zero because it contains the factors that ensure it is a counter example to the last Fermat theorem. As such these two factors are always a pair of divisors of zero and as such if you multiply them you get zero. I did not give it a second thought, only remarked the determinant is zero and as such this likely spells some trouble for such a Frey elliptic curve. Recall that if the determinant is zero, there is at least one double zero to be found in your equation. Two weeks back I looked at it with a bit more detail and the result was rather surprising: The cubic part of such a Frey curve indeed gets some double zero, but all in all in the space of integers modulo 35 the thing has four zero’s! But all these Frey curves go like
y^2 = x(x + A)(x – B) so the ‘x-part’ is a degree 3 polynomial. And a 3 degree polynomial has at most 3 zero’s, or not?

Wrong! A 3 degree polynomial has at most 3 zero’s (or precisely 3 zero’s if you count with multiplicity) but in spaces with divisors of zero it is possible to get extra zero’s. On the spaces of 3D complex and circular numbers it is easy to find a parabola with 3 zero’s. Take p(x) = x^2 – x. If you solve for p(x) = 0 you basically try to find numbers that are their own square. After all if a number is it’s own square, there is no denying that
x^2 – x = 0. Of course x = 0 and x = 1 are their own square, but on the three dimensional complex and circular numbers the center of the exponential circle has also that impotant propery. The center is usually denoted as the number alpha and yes that is a zero too. Yet you cannot use such an extra zero in a factorization of such a parabola. The factorization is and stays
x(x – 1) and if we substitute alpha in we get alpha(alpha – 1). But ha ha ha these last two factors are a pair of divisors of zero and that is why it’s zero…

I also found a cute applet that you can use for making graphs of elliptic curves modulo a single prime number. Most of the time I needed modulo 35 or anything with two prime numbers in it and the applet tries to warn me every time when I do that. I also made a few graphs from counter examples to the last theorem of Pierre de Fermat with it. That was relatively funny and made me decide to write this post. Of course we do not need some applet to validate my counter examples to the last theorem because there is plenty of proof. But it was funny anyway.

Links to the two relevant applets:
Elliptic curves over finite fields, and
Curves over finite fields. With the last applet you can make those counter examples to the last theorem of Pierre de Fermat because it allows you to use exponential expressions. It has to be remarked that the applet is not so good at exponential series so already at the stuff modulo 35 it runs awry and returns gibberish instead of a nice flat zero.

This post grew longer as planned before hand, but that is with most posts I write. You always think ‘ah that is simple to explain in just a few words’ but if you try that and also estimate the math content of the average math loving person, I always need more and more words to explain the stuff involved… Well so be it.

This post is 9 pictures long in the format or size I use lately: 550×825 pixels.

From modulo 35 and higher the applet starts disfunctioning.

Ok, let’s hope there are no unseen typo’s in the above. I always hate when later I read stuff already published to the internet and find out there are typo’s in it…
The next post is likely an infinite list of counter examples to the last theorem of Pierre de Fermat or it will be about a cute antenna design based on the 3D exponential circles. This cute antenna should produce circular radiation and as far as I know this design is not used anywhere. (I do not know if it is better compared to what we have at present date, after all antenna design is a well developed technical field.)

Ok, that was it for this post.

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.