# Counter examples to the last theorem of Fermat using the number 210.

Ok ok one more post upon the easy to find counter examples to the last theorem of Fermat. In this post we will take a look at the real integers modulo 15 and modulo 210. It still amazes me how easy it is to find counter examples to the last Fermat theorem using the integers modulo n where n has at least two prime factors. From my own education I remember that the integers modulo n are studied in math mostly via additive groups and multiplicative groups. For some strange reason it is not commonly studied via rings where you have the benefit of addition and multiplication inside one simple to understand structure of numbers… Inside professional math there is always that tendency to study fields only, of course there a legitimate reasons for that like it makes math life often more simple. But rings are not fields, rings allow for non zero numbers that are non-invertible anyway. As such you can always find plenty of pairs of so called ‘divisors of zero’ and once you have stuff like that it is always a piece of cake to find counter examples to the last theorem of Fermat.

Yet I tried a few times to find some counter examples on the internet but all I got was boatload after boatload of total nonsense like the weird stuff paraded in the previous post. Could it be that math professors tried to find counter examples to the last theorem of Fermat while they never dipped into the power of the divisors of zero? That’s crazy because the Fermat theorem was open for about 350 years. I think many people have found the easy to understand results in this post before I did but if they tried to get the stuff out they were blocked by the scientists of those days and as such in the year 2021 it is hard to find something back.

Compare it to electron spin; it is hard to swallow that I am the very first person in history that claims electrons cannot be magnetic dipoles because it is just not logical for hundreds of reasons. Yet in the daily practice of how science is done at the universities, it is a no show that electrons are magnetic monopoles. What happened to all those other persons that understood that electrons cannot be magnetic dipoles? Well at least they got neglected and university life just went on with electrons being a magnetic dipole because ‘we are so smart’ and ‘the standard model explains almost everything’. And more of that nonsense…

This post is 8 pictures long, all of the usual size of 550X775 pixels.
Since it is about counter examples to the last Fermat theorem I expect it will not make much headlines in the news for another 3500 years.
After all the only thing university people are good at is being incompetent…;)
Here we go:

At last I found a more or less readable article about near misses of the last Fermat theorem. It was found inside old work from Ramanujan so that is always interesting. Most of the time when I looked for counter example to the last Fermat theorem I only find piles of garbage but this time I tried it with Duckduckgo and something readable comes floating up:
Ramanujan surprises again.
https://plus.maths.org/content/ramanujan

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

# Why can’t I find counter examples to Fermat’s last theorem on the internet?

After a few weeks it is finally dawning on me that it might very well be possible that the professional math people just do not have a clue about how easy it is to find counter examples to the FLT. (FLT = Fermat’s Last Theorem.) That is hard to digest because it is so utterly simple to do and understand on those rings of integers modulo n.
But I did not search long and deep and I skipped places like the preprint archive and only used a bit of the Google thing. And if you use the Google thing of course you get more results from extravert people. That skews the results of course because for extraverts talking is much more important compared to the content of what you are talking or communicating. That is the problem with extraverts; they might be highly social but they pay a severe price for that: their thinking will always be shallow and never some stuff deeply thought through…

As far as I know rings of the integers modulo n are not studied very much. Of course the additive groups modulo n are studied and the multiplicative groups modulo n are studied but when it comes to rings all of a sudden it is silent always everywhere. And now I am looking at it myself I am surprised how much similarity there is between those kind of rings and the 3D complex & circular numbers. Of course they are very different objects of study but you can all chop them in two parts: The numbers that are invertible versus the set of non-invertibles. For example in the ring of integers modulo 15 the prime factors of 15 are 3 and 5. And those prime factors are the non-invertibles inside this ring. This has all kinds of interesting math results, for example take the (exponential) orbit of 3. That is the sequence of powers of 3 like in: 3, 3^2 = 9, 3^3 = 27 = 12 (mod 15), 3^4 = 36 = 6 (mod 15) and 3^5 = 18 = 3. As you see this orbit avoids the number 1 because if it would pass through 1 you would have found an inverse of 3 inside our ring and that is not possible because 3 is a non invertible number…

Likely my next post will be about such stuff, I am still a bit hesitant about it because it is all so utterly simple but you must never underestimate how dumb the overpaid math professors can be: Just neglecting rings modulo n could very well be a common thing over there while in the meantime they try to act as a high IQ person by stating ‘We are doing the Langlands program’ & and more of that advanced blah blah blah.
Anyway it is getting late at night so from all that nonsense weird stuff you can find on Google by searching for counter examples to the last theorem of Fermat I crafted 3 pictures. Here is the first one:

I found this retarded question on quora. For me it is hard to process what the person asking this question was actually thinking. Why would the 2.999…. be important? What is this person thinking? Does he have integer solutions to say 2.9 and 2.99 and is this person wondering what would happen if you apply those integer solutions to 2.99999999…..???????

It is retarded, or shallow, on all levels possible. So to honor the math skills of the average human let’s make a new picture of this nonsense:

We will never be intimidated by the stupidity of such questions and simply observe these are our fellow human beings. And if ok, if you are a human being running into tons of problems, in the end you can always wonder ‘Am I a problem myself because I am so stupid?’

If you have figured out that question, you are getting more solid & you look more like a little cube:

I want to end this post on a positive note: Once you understand how stupid humans are you must not view that as a negative. On the contrary, that shows there is room for improvement.

# The last Fermat theorem (positive version) versus the number 1.

This is a short post; just over 3 pictures long. We make a few calculation on the ring of integers modulo 35. Of course that is a ring and not a field because 35 has two prime factors namely p = 5 and q = 7. These two prime factors form so called divisors of zero, that means that pq = 35 = 0 inside the ring of integers modulo 35.
Because the two prime factors have this property, that has all kinds of simplications when it comes to expanding (p + q)^n inside this ring. That is what I name the ‘positive version’ of the last theorem of Fermat: The ring of integers modulo 35 is a simple number space where the last theorem of Fermat is possible, here we again have 12^n = (5 + 7)^n = 5^n + 7^n.

In this post I use the fact that the prime numbers 5 and 7 are also relatively prime and as such you can make a linear combination of them to get the number 1. And once you have the number 1 you can use them as a basis for the entire ring of integers modulo 35. But if you have a healthy brain, likely you will remark that it is far more easy to just use the counting numbers 1, …, 35 or just 1 to craft such a basis… So I understand that you might think I am crazy to the bone. Of course I am crazy to the bone but there is a goal in this utter madness. Take for example 3*5 – 2*7 = 1, this is one possibility to form the number 1 as a linear combination of 5 and 7. Since both terms contain one of the pairs of divisors of zero as a factor, this linear combination allows for a positive last theorem of Fermat decomposition: For a natural numbers n we have that: (15 – 14)^n = 15^n + (-1)^n*14^n = 1.
Although such expressions are very cute looking, it has no significant math depth anyway. All in all this post it totally unimportant because it is all so simple. The post upon the 3D Gaussian integers is far more important because there it was possible to write the number 3 as a linear combination of two 3D Gaussian integers. As such for the first time in about 350 years it was the first serious counter example against the last theorem of Fermat because that number 3 was just on the line of integers. It was not something inside some modulo number space or so, that was the real deal for the first time in 350 years.

Will math professors react on such a finding? Of course not. For example they would reason before the finding that if you can’t use 3D complex numbers to find only one significant result in algebra or number theory, that proves 3D complex numbers are useless.
And after the counter example to the last fermat theorem? Well math professors are the most smart people on earth, they are higly agile and adeptable and now the reasoning will likely be something like: In the entire history of mathematics nobody has ever used 3D Gaussian integers. This all is so far fetched that this is not serious math

Well that is how they are and there is no changing that kind of behavior I just guess. Anyway enough of the blah blah blah. The post is just over 3 pictures long, has no mathematical significance anyway and I hope you have some fun reading it.

It is now one hour after mdinight so it is time to hit that button named ‘Publish website’. Live well & think well my dear reader. See you in the next post or so.

# What is one-way light speed? + A plasma lamp in a magnetic field.

Lately the Veritasium guy from the Youtube channel with that name came out with a video that made me think. It seems that the only way light speed has been measured experimentally is by using a mirror and as such you always measure a so called ‘two-way light speed’ average. It is possible that in this universe light has some preferred direction and in that derection it goes faster compared to say the opposite way.

You might wonder as why that is but that is the old problem Mr. Einstein faced & solved: It is very hard to get two clocks synchronized when they are apart. Vertitasium explains that if light has a preferred direction, in that case it all gets even harder and rather complicated.

Anyway to make a long story short, he also claims there has never been such a ‘one-way’ experiment. That made me think about it and I think that I have found a solution that does not depend on the nasty synchronization problem. All you need is two atomic clocks that are always a fixed distance apart. It gets more complicated compared to where there is perfect sync but if the universe has a preferred direction for light to go in, from the data you collect you should be able to find it.

First let me show you via a screen shot from the video what the usual way is to experimentally measure the speed of light:

In the next six pictures I try to explain that using two atomic clocks on two satellites can pull the trick off. Both satellites have a laser or for that matter any em radiation would do like normal radio waves. And on both satellites there is perfect registration of the local time of the times a laser signal was put out or received. Both satellites should be in the same strength of the local gravity field so that their atomic clocks run at equal speeds. Every day both satellites send out one laser pulse on a fixed time and on a daily basis these send & receiving times are recorded.

We proceed with a video from the Brainiac channel. On Youtube there are more than one Brainac channels, I mean the guy with the big magnets. And with big I really mean big, the most heavy ones are a staggering 13 kg. A couple of years ago I wondered if I should buy a plasma lamp in order to study how my own set of small neodymium magnets should influence the plasma. These lamps cost only 20€ so that was not the problem. But I have already plenty of lamps so I decided not to do it. After all the photo’s from how an old television set reacts on the neodymium magnets should be enough.

Anyway that is what I thought: Given the fact the audience is composed of scientists, simply communicating the facts should be enough. And applying a bit of logic simply says electrons are not magnetic dipoles but in order to explain the behavior on an old television set is far better explained by electrons being magnetic monopoles. Of course now we are five or six years further down the timeline all I observed is that university people are still very good at just one thing: being important. And no no no, of course we do not talk about that. Electrons magnetic monopoles? Great minds from physics, also people who were very important, said it ain’t so. So no no no, we are important and those crazy people from outside science should shut up and pay the taxes we need for being so important.

Yet now a few years later the video from the Brainiac guy shows that the plasma in the plasma lamp does not react at all as you might expect from a bunch of magnetic monopoles. On the contrary: If he applies one of those 13 kg neodymium magnets, the plasma lamp stops working. So I am glad I never bought one of those lamps because that would make me doubt my own insights that were derived from the electron cannons in the old television set… But the Brainiac guy has much more electronic equipment and he soon found out that the plasma in a plasma lamp is steered by and alternating electric current. So the plasma shakes hin und her with a high frequency and it is not streaming in one direction or so…

Furthermore he explains perfectly why the lamp stops working: the applied magnetic field from the 13 kg magnet stops the transformator in the plasma lamp from working properly. And that explains why the lamp stops working… By the way, if electrons were really magnetic dipoles, the 13 kg magnet would never hinder the functioning of a transformator because a magnetic dipole the size of an electron is by definition neutral for external magnetic fields.

Well here is the perfect video if you want to understand a bit of nature and in case you are one of those fake scientist only occupied with the importance of self, why not walk to a mirror so you can look at yourself?

This post is getting far to long for the attention span people have in the present media environment. But I want to show you also a part that the Brainiac guy does not understand: the next screen shot from his video shows only electrons that are repelled by the giant magnet he uses. If he would have used an other direction you could have seen also the electrons that are attracted by the magnet. All of this stuff nicely confirming year in year out that it is impossible for electrons to be magnetic dipoles…

Ok, end of this far too long post. See you in the next post & thanks for having a long enough attention span for reading these very words.

# Funny format/more pics needed & idiots at MIT observed?

Slowly but surely I am getting better at the GIMP (a free program for manupilating images). Right now I can place pictures in perspective while the GIMP als has a 3D picture manupilating tool I haven’t even used by now. Now one day I will have to end my usual way of formatting pictures, the biggest disadvantage is that you must always have a windows XP computer. May be it is possible to run a virtual XP on a windows 10 system but I never managed to get it properly at work. On the other hand computers in for example pin automats (money machines at the bank) still seem to work on XP. So likely in the future their will still be motherboards and CPU’s that allow for a fresh install of that mighty windows XP system.

Anyway with GIMP you can easily use the perspectives tool and place rectangular selections into a perspective like shown below:

No, the above format for publishing math does not work properly I guess. It is a screenshot from a post from earlier this year: Calculating the 3D exponential circle using first principles.

Now in another development I was also not very lucky. I found a few pictures of the creation of an electron-positron pair in a bubble chamber. Now if my view on electrons and positrons being magnetic monopoles is correct and because in a bubble chamber you have a magnetic field present, from the moment of creation they should start accelerate in opposite directions. And I thought all I needed was just one Google picture search but the results were a bit disappointing. Yes you can find some pictures but most of the time it is just one photo that is recycled over and over. Another disadvantage is that you see the electrons and positron bubble paths only in the direction of the magnetic field that is applied so that the electrons & positrons can do their typical circular movement due to the Lorentz force so it is abosolutely not possible to see the eventurental acceleration into the direction of the magnetic field lines… Well most of the time you find the next picture and yes it looks like the one particle is ‘going in’ and the other is ‘going out’ but that is all there is. No sideviews found at all and that is what I need. An interesting phenomenum that should occure is the next: Due to the bubbles there has to be some kind of drag on the electron and positron. So their velocities along the magnetic field lines should take on some limiting value. If that can be found that alone should be enough to validate that electrons canny magnetic charge and that all this ‘tiny magnet’ stuff is total bs.

And the last item for this post is the MIT people. Again it is blah blah blah because we now have stronger magnets we can make smaller nuclear fusion reactors. But if my view of electrons being magnetic monopoles in the end will be victorious, stronger magnetic fields do not solve the acceleration problem. Electrons get constantly accelerated and because there are two types of electrons namely the north and south charge they will get accelerated into opposite directions.

I have been saying this for years and years and still the university people keep on doing their retarded thing and not proves that electrons are actually magnetic dipoles. In the meantime those imcompetent shitholes keep on making promesis for a better future when it comes to energy for the population and blah blah this & blah blah that.

Remember the time that Lockheed Martin came out with the same kind of bullshit? By now we should have had the first mobile fusion reactors and of course they are nowhere to be found. And now we have exactly the same nonsense from MIT.

It’s not going to work, but try explaining that to a bunch of total incompetents! Here is a Youtube with the MIT stuff (about six minutes long):

We are dealing with a bunch of people too stupid to find out in centuries of time how 3D complex numbers should be found (or defined). And all I get is total neglect and they go on with their blah blah blah. Give us, the tax payer, finally some fucking proof that electrons are magnetic dipoles and that the structural instability of the plasma is not caused by accelerated electrons! Of course, as usual, there will be silence. Only the sound of silence combined with blah blah like ‘we now have stronger magnets’. Climate change is not going away in the meantime and it is charlatans like this that will make people going on with polluting the atmosphere more and more because there is some false hope nuclear fusion will save the day. Once more: Likely it is not going to happen. Look at the Lockheed Martin folks; they still have nothing to show for despite their past blah blah blah about having stronger magnets…

Ok, that was it for this post. The next post is about a math article from the preprint archive that is about 3D complex numbers. So keep tuned and see you next time.

# Impending Nobel prize & recycled Pythagoras theorem & it’s ‘inverse’.

Tomorrow is the new Noble prize in physics out, actually it is already past midnight as I type these words so it is actually today. But anyway. I am very curious if this year 2020 the Nobel prize in physics will once more go to what I name those ‘electron idiots’. An electron idiot is a person that just keeps on telling that electrons are magnetic dipoles because of something retarded like the Pauli matrices. May be idiot is a too harsh word, I think that a lot of that kind of behavior or ideas that can’t be true simply stay inside science because people want to belong to a group. In this case if you tell the official wisdom of electron spin you simply show that you belong to the group of physics people. And because people want to belong to a particular group they often show conformistic behavior, when it comes to that there is very little difference between a science like physics or your run of the mill religion.

In this post I would like to share a simple experiment that every body can do, it does not blow off one of your arms it is totally safe, and shows that those Pauli matrices are a very weird pipe dream. Here we go:

The official explanation of the Stren Gerlach experiment always contains the next: If electron spin is measured into a particular direction, say the vertical direction, if later you measure it again in a direction perpendicular on the vertical once more it has 50/50 probability. So if it is measured vertically and say it was spin up, if you after that measure it in say a horzontal manner once more the beam should split according to the 50/50 rule.

Ok, the above sound like highly IQ level based on lots of repeated laboratorium experiments. Or not? And what is a measurement? A measurement is simply the application of a magnetic field and look what the electron does; does it go this way or that way?

Electron pairs are always made up of electrons having opposite spins, in chemistry a pair of equal spins is named a non-bondig or an anti-bonding pair. Chemical bonds based on electron pairs cannot form if the electrons have the same spin.

Now grab a strong magnet, say one of those strong neodymium magnets and place it next to your arm. Quickly turn the magnet 90 degrees or turn your arm 90 degrees, what does happen? Of course ‘nothing happens’ but if electron spin would follow that 50/50 rule, in that case 50% of your electron pairs would become an anti bonding pair. As such your flesh and bones whould fly apart…

Now does that happen? Nope njet & nada. As far as I know it has never been observed that only one electron pair became an anti-bonding pair by a simply change of some applied external magnetic field…

As far as I know the above is the most easy day to day experiment that you can do in order to show that electrons simply do not change spin when a different magnetic field is applied…

I have been saying this for over five years but as usual when it comes to university people there is not much of a response. In that regard physics is just like the science of math: It has lost the self cleaning mechanisms that worked in the past but now in 2020 and further those self cleaning mechanisms do not work anymore. It is just nothing. It is just a bunch of people from blah blah land. So let’s wait & see if one of those ‘electron idiots’ will get the Nobel prize tomorrow.

Luckily I have a brain for myself. I am not claiming I am very smart, ok may be compared to other humans I do well but on the scale of things like understanding the universe I am rather humble. I know 24/7 that a human brain is a low IQ thing, but just like all other monkeys it is the only thing we have.

Very seldom the human brain flares up with a more or less bright idea that simplifies a lot of stuff. A long time ago I wanted to understand the general theorem of Pythagoras, I knew of some kind of proof but I did not understand that proof. It used matrices and indeed the proof worked towards an end conclusion but it was not written down in a transparent way and I just could not grasp what the fundamental idea’s were.

So I made a proof for myself, after all inside math the general theorem of Pythagoras is more or less the most imporatant theorem there is. I found a way to use natural induction. When using natural induction you must first prove that ‘something’ is true for some value for n, say n = 2 for the two dimensional theorem of Pythagoras. You must also prove that if it holds for a particular value of n, it is also true for n + 1. That is a rather powerful way to prove some kind of statement, like the general theorem of Pythagoras, holds for all n that is holds in all dimensions.

I crafted a few pictures about my old work, here they are.

It is from March 2018 when I wrote down the ‘inverse’ theorem of Pythagoras:

And from March 2017 when I wrote the last piece into the general theorem of Pythagoras:

Ok, let me leave it with that and in about 10 hours of time we can observe if another ‘electron idiot’ will win the 2020 Nobel prize in the science of physics. Till a future post my dear reader. Live well and think well.

# Two video’s to kill the time.

Two very different subjects: the earth magnetic field and the standupmath guy has a great video about the perimeter of an ellips.

Video 1) From the Youtube channel Scishow a video with the title
‘Satellite Squad Goals: The Cluster Mission to the Magnetic Field’.
For me that video contains relatively much completely new stuff, the fact that there are 4 satellites out there constantly monitoring the earth magnetic field was unknown to me.
And the presenter of the video claims that after the so called ‘magnetic reconnection’ the charged particles from the solar wind slam into the north & south pole of the earth with a staggering 10 thousand km/sec. I did not know it was that fast…
The official explanation for the acceleration of for example single electrons is that you must have an inhomogeneous magnetic field. After all these folks think that electrons have two magnetic poles and if the electron goes through a magnetic field that varies in space the two forces on the north and south pole of the electron do not cancel out and there is a net force responsible for the acceleration. There is only one problem: they simply multiply the electron magnetic moment against the gradient of the magnetic field and voila: that’s it. But if the acceleration is explained as a difference in opposing forces, should you not take into consideration the size of the electron? Yes of course, but since physics professors are so terribly smart why don’t they do this? Well if you take the size of the electron into your calculations, there is no acceleration or better it is basically zero.

Now years ago I tried to estimate how stong a magnetic field had to be to accelerate one of those dipole electrons with a acceleration of only 1 meter per second squared. If memory serves I used an ‘electron size’ of 10 to the power -15 meter (in reality it is even much smaller) and again if memory serves you needed magnetic fields with a gradient of over 100 thousand Tesla per meter.
And if you think about that estimation it makes a lot of sense: electrons are very small and as such have an extreme density given their size and mass. Say it is in the order of the density of a neutron star. And if you try something with the density of a neutron star to accelerate with the difference of a magnetic field, likely you won’t go far…

Ok, suppose for the moment that the electrons are the long sought magnetic monopoles. So they are not magnetic dipoles but the electrons themselves are magnetic monopoles just like they are electric monopoles.
Now look at the picture below: it is about when the magnetic reconnetion just closed. Just before the closing along the magnetic field lines emergin from the earth north & south pole, the particles were expelled because they carry the wrong magnetic charge. But when reconnection takes place, the particles that were expelled by say the earth south pole find themselves back on a trajectory going to the earth north pole. And as such they will get accelerated into that direction.

Yet a couple of years ago when I published those estimations that show you need crazy gradients for all that shit to be true, of course nobody reacted. All those university professors in physics, when you tell them that extra ordinary claims like the electron being a magnetic dipole also needs extra ordinary proof, all of a sudden they are deaf deaf deaf.
These people they don’t have any experimental proof that the electron is a magnetic dipole. And worst of all: They don’t even think about it…
Finally, here is the SciShow video:

Video 2) From the Standupmath guy a video about the perimeter of an ellipse. Weirdly enough it is not possible to find a more or less simple expression for the perimeter of an ellipse. Of course a long long time ago I tried to find an expression myself but using the standard stuff like arc length brings very fast a lot of headache. With the present day of math tools it is completely not possible to derive a good expression for the perimeter of an ellipse.
What I did not know is that there is a world of approximation stuff out there for estimation such ellipse perimeters. And of course in itself this has it’s own logic: after all an ellipse is more or less completely defined by saying what it’s two half axes a and b are. You can always fix one of those axis to 1 say b = 1 and study the perimeter problem as a function of the variable a. You do some curve estimation, you drink a few pints of beer and later when you are sober again you drink some green tea.
And you conclude some curve estimation is relatively good but that all in all the ellipse perimeter problem is just too large for our human brains that in general are not good at doing math.
There is only one exeception; Ramanujan.
In the next picture you see one of those Ramanujan approximations and once more you see how the human mind should work if we were living in a better world:

The video is here, 21 minutes long but worth the time:

Ok, that was it for this post. Think well, live healty and try to make some bio fuel from the basic ingredient known as ‘math professor’.
In that case we will find ourselves back in a better world, or not?

# On a simple yet curious integral identity.

In a pile of paper notes I found back this curious identiy, shall I throw it away or write a small post upon it? Most things I throw away, if I would write posts about everything that comes along this website would be 1340 posts long…

I found it in a video from Presh Talwalkar, Presh runs the video channel ‘Mind your decisions’ on Youtube. There is only a tiny problem: I can’t find back the original video. And since Presh has posted about 518 video’s it would take a long long time to find that video back. So no video included.

Anyway the video started more or less like next: Presh throws in three difficult looking integrals and asks his viewers to take five minutes and try out if they can find the answer. It looks like those integrals are for relatively fresh students and I was just like ‘you can’t ask such integrals for starting students!’ But likely those students had seen this identity and as such those nasty looking integrals could be solved with two fingers in the nose if they just recognized it to be this curious identity…

By the way, Presh his channel has about 1.4 million subscribers. My applause goes to Presh. One point four million is not a bad result, for example the university sponsored channel Numberphile has over three million subscribers so on his own Presh is doing just fine.

So this post is not about 3D numbers, complex or circular but upon this identity. It is only three pictures long so it won’t take much of your precious time. Let’s go:

Of course with symmetric I mean a function that is even with respect to the midpoint of the interval [a, b]. Let’s try if we can post a link to the Presh Youtube channel: Mind your decisions.

Ok, that was it for this post. No idea yet what the next post is about, after all most things I just throw away. So till updates my dear reader.

# Teaser picture for the next post.

After a lot of rainy days it was perfect weather today for the time of the year. It has been 3 weeks already since the last post and it is not that I have been doing nothing but the next post still isn’t finished. I told you that we would be looking at a parametrization that solves all 5 equations from the last post. So let me give you the parametrization in the teaser picture below. I also included the parametrization based on the modified Dirichlet kernels, by all standards the discovery of those modified kernels was one of the biggest discoveries in my study of higher dimensional number systems. To be precise: I found the first modified Dirichlet kernel years ago when I studied the 5D complex space.

In the last post I may have sounded a bit emotional but that is not the case. I am more or less one 100% through with the behavior of the so called math professors. They are incompetent to the bone and although that is not an emotional thing, it is that coward behavior that I do not like in those people. No, if it is highly overpaid, utterly incompetent and on top of that day in day out a coward, better show them the middle finger.

After having said that (I wasn’t expecting an invitation anyway) let’s look at the teaser picture because it is amazing stuff. I remember when I wrote down the parametrization for the very first time. At the time I did not know if the cosine thing would work because say for yourself: if you have a periodic function and you make two time lags of it, how likely is it they will form a flat circle in 3D space? But the cosine together with the two time lags does the trick because it is not hard to prove the parametrization lies in the plane with x + y + z = 1.

Ok, here is the cute parametrization for the 3D exponential circle:

I think next week everything is ready so likely I can finally upload the next post. So thanks for your attention and till updates.

# On Benford’s law.

Benford’s law is a statistical observation. If I remember the story more or less correct, Benford found at some point in time that those old logarithm tables had pages that were far more worn out compared to others. And it seems that people used those old (but at the time very important) tables much more for numbers starting with small digits like 1 or 2 and much less for high leading digits like 8 or 9. The observation was that the probability of a leading digit of d was given by log(1 + 1/d). I remember that during a train ride to the city of Utrecht about two decades ago I found a very simple distribution that gives the Benford law perfectly for numbers written in the usual base 10. Basically if you use a uniform distribution in the exponent, that more or less always gives rise to some approximation of Benford’s law.

A few weeks back for no reason at all, I did a search on the preprint archive on the subject of Benford’s law and a rather strange article popped up. It is written by Kazifumi Ozawa. Title: Continuous Distributions on $(0, \infty)$ Giving Benford’s Law Exactly.