General Theory Part 4: The Fundamental Theorem of Calculus.

4D complex numbers, CR equations, Integration

It is already some time ago that I posted the last part of the new ‘General Theory’ series. That was on Oct 28 in 2023 and the last part was about the Cauchy-Riemann equations for n-dimensional real space aka the n-dimensional complex and circular numbers. Since the CR equations govern anything related to differentiation, how to calculate partial differentials and how they relate to each other and most of all: how to find the derivative of a function, you will need that to understand this post. This post is the fundamental theorem of calculus and that is of course integration along a contour of as I often name it a bit wrong ‘line integration’.
Basically the result is the same in all spaces: Under certain mild conditions you can integrate along a contour some function that we name f. And if you do that the result is the difference of some primitive lets name it F in the end points of that curve of integration.
Now in the last post on integration where we looked at the fundamental theorem of calculus for the 3D complex numbers I pointed at the fact it is now the year 2025 and why did it take so long? Well all those years I assumed that getting a primitive F of some function f was not always assured especially when you were integrating in the sub spaces of non invertible numbers that higher dimensional numbers have. And yes there are a boatload of troubles there, sometimes it is very hard to find a primitive and it is not always clear if you will actually find it.

So in this post I turned the troubles upside down and I simply start with a primitive and it is proven that the integral of the derivative of this primitive is the primitive. To put it simple:
1) I have some primitive F or F(X),
2) I differentiate it to get some f or f(X) and
3) I integrate along a curve from A to B and guess what?
4) I get F(B) – F(A).

But by doing that, how to find the derivative inside the sub space of non-invertible numbers? After all you can’t devide by then so how to even make sense of the most basic limit for calculating the derivative? From the ‘math technical’ side of things, this was easy to solve with just a tiny modification this division by a non-invertible number can be avoided. With that this approach of starting with a primitive looks like a clean approach without much unsolved problems of understanding.
Weirdly enough it was a easy ‘do-able’ so by just differentiating the set of all primitive functions I could avoid proving the existence of primitives when you take the integral of some function f.

The post is six images long and have size 550×1500 pixels, there are an additional two images in the appendix where I work out the primitive of ZdZ in the space of 4D complex numbers.

So lets recap what we have done in this post: In case it is correct that this universe is 13.8 billion years old, in that case the math professors have found only three versions of the fundamental theorem of calculus. They have two slightly different versions for the real line and one for the complex plane. So basically they only have two Fundamental Theorems of Calculus.
In this post we looked at the n-dimensional real spaces equipped with either the complex or the circular multiplication. So for every fixed dimension n there are two fundamental theorems related to those two multiplications. As such in this post we found an infinite number of fundamental theorems so although I don’t like it to write general theory, all in all this was not a bad day at the office.

Because I never made a so called ‘category’ of general theory on this website you must use the search function or use the next internal links:
Proof that Z^2 = -1 cannot be solved on real spaces with an odd dimension. (General theory part 1.)

General theory Part 2: On a matrix named big E.

General Theory Part 3: Cauchy-Riemann equations.

So for the time being included this post General Theory Part 4 on integration, for the time being, this is all general theory that there is. It contains the ancient quest for a number who’s square equals minus one. The matrix big E is handy if you want to study all kinds of multiplications while part 3 and 4 cover differentiation and integration.


Comparison: Einstein de Haas effect versus the Stern Gerlach experiment.

Magnetism

This post is basically a video about the so called Einstein de Haas effect from the Action Lab (a video channel). This experiment is often mentioned as experimental validation that electrons have so called “Intrinsic angular momentum”. The experimental setup is very easy to explain, look in the next picture:

A metal cylinder is hanging from a wire (the guy in the video uses tooth floss because that has no winding twist in it so the cylinder will not rotate). If placed in a vertical magnetic field or such a vertical magnetic field is flipped on, the cylinder rotates a little bit.
Often in the experimental setup a coil magnet is used, but it can be any more or less perfect vertical magnetic field. The effect of the rotation is rather small so in video’s like this you often see some shiny or reflecting metal glued to the cylinder and with a light the rotation is amplified for us to see.

This is all there is; a relatively simple experimental setup.

The explanation you always see is that the unpaired electrons in the metal cylinder align themselves with the applied magnetic field. So that’s the logic for the explanation of this experiment.

How different is it for the Stern Gerlach experiment that is very similar because it is about the behavior of unpaired electrons in a vertical applied magnetic field. In the SG experiment the beam of silver atoms is split in two and now the logic is as next, quote:

This means that when you take a beam of electrons whose angular momenta are all randomly oriented, if you measure the z component of angular momentum you get one of only two different values.

But if your explanation in the Einstein de Haas effect would be this 50/50 percent probability in spin up or spin down, that would imply zero rotation and therefore once more day the physics professors will talk out of their necks and now it must be logical that all electrons align.
Source of my above quote:
Measuring Electron Spin- the Stern-Gerlach Experiment

Furthermore it is a fundamental basis of quantum computing that it must be possible to have superpositions of quantum states. So if the explanation of the Einstein de Haas effect would be correct, there is no randomness in electron spin measurements via application of external magnetic fields. You just can’t eat it from both sides: either all electrons will do the same or the probabilistic nature of quantum mechanics is true. Anyway here is the video with the title: Do Electrons Really Have “Intrinsic” Angular Momentum?

https://www.youtube.com/watch?v=uQ5w4_0S2l4

That was it for this post on magnetism in particular the very different explanations you hear when we are talking about the same thing: The reaction of an electron on a vertical magnetic field.

The Fundamental Theorem of Calculus for the 3D Complex Numbers.

3D complex numbers, Integration

Some time after I started writing this post I thought “Hey why not look up what is more or less offically said on this theorem in the internet”. So I did and found out there are three versions of this theorem that basically says you can do integration with a primitive. That was a tiny surprise to me because the way I remembered it was there is just only one. There are two versions on the real line and one for the complex plane. The first real line version says that the integral of a function on the real line equals the difference that the primitive has in the begin and endpoint of integration.
The second real line version uses a variable in the endpoint of integration, say x, and define a primitive F(x) of a function f(x) as such. After that it must be shown that the derivative F'(x) = f(x). And the third version says you can do line integration (or integration along a curve traditionally named gamma) also using a primitive but now you must take into account the way differentiation works in the complex plane.

So there is not one such fundamental theorem but the official theory says it’s three. Now why three? Very simple: The professional math people know of no other spaces where you can do integration with a primitive. I’ve said it before and repeat it once more: The 4D quaternions are nice things but when it comes to differentiation and integration it is hard to get a bigger mess of total gibberish. That’s why the professional math professors don’t have such a fundamental theorem for the quaternions.

But for the 3D complex numbers that are the main topic of this website, it can also be done. But hey this is now the year 2025 and this website is almost 10 years old and on top of that I found the 3D complex numbers back in the year 1990, so why only now this theorem in the year 2025?
Now over the years I have always used this kind of integration when I needed it. For example the number tau for the 3D complex numbers was calculated the first time by using integration while I developed the matrix diagonalization methods only later to deal with the problems you get in say five or seven dimensional complex numbers.
For people who don’t have it clear what the numbers tau are: They are the logarithm of an imaginary unit. For example the log of the imaginary unit i on the complex plane is i times pi/2 as was already found by the good old Euler. Now for the 3D complex numbers it’s a bit more difficult but you can find such log’s of imaginary numbers indeed with integrating just the inverse.

But I always thought there would be some kind of trouble if you integrate just inside the subspaces of non-invertible numbers. So that’s more or less why I never ever formulated such a fundamental theorem in all those years. I only used integration when I needed it and that was it.

To my excuse there are indeed some subtilities, I once tried to find the primitive of say e^X and yes, no problemo, it is just e^X. While if you calculate the integral now with a 3D number X but multiply it by that famous number alpha, the primitive changes in a dramatic fashion.
So all those years I thought that even for say the exponential function there was not just one primitive to do all the work. Yet now I take a deeper look into it, this was all a bit stupid of me.
So it’s a lame excuse but compared to the professional math professors who can’t even find the complex 3D numbers, I shine as the stable genius I am… Ahum, this post is only 9 pictures long and has 3 additional figures and one video about the fundamental theorem on the real line. So all in all there are 12 pictures and 1 video below.

Lets get this party started:

The next picture is the so called Figure 1 picture and it shows where the determinant is one. So on the red colored graph you can do the ‘divide by beta’ thing in the limit for the derivative of a function. The problems with taking such a limit on the space where det(X) = 0, you can’t divide by such a beta so doesn’t that cause some problems? Well no, you can always flip hin und her between the two above definitions of taking a derivative.

Figure 1: I am not crazy: Can’t divide by beta? Well, try a multiplication by beta…

And here is the so called Figure 2 picture where I depicted a line segment inside the main plane of non-invertible numbers in the 3D complex numbers. Therefore I invite you to think a bit along those lines, does it matter if they are inside or outside the space of det(X) = 0?

I have a link for you to a page from the website from Stephen Wolfram where the three fundamental theorems of calculus are explained.
Fundamental Theorems of Calculus
Now I published the above about 24 hours ago but I was forgotten to place the link to Wolfram. And today I was watching youtube and to my surprise a video from Hannah Fry came floating along while it said it was about the fundamental theorem of calculus. It’s all very very basic because Hannah uses it also to craft an introduction to integration using the limit of an elementary Riemann sum. For most readers it is a bit too simple I guess but in case you harldy know what integration is, for those it is a very good video.

And for no reason at all I also made a cube with her face on it. We all love Hannah because she is relatively good at popularizing math. And that’s a good thing because it makes the general population a bit less stupid. Anyway that is what you might hope for but don’t let you hope become to big because the human brain and math is often not a good combination. We’re just a fucking stupid monkey species, ok we are the smartest monkeys around but we’re still a monkey species…

This is the end of this post, now we have a fourth so called Fundamental theorem of calculus. Lets leave it with that.

Two more video’s on the Stern Gerlach experiment from the year 1922.

Magnetism, Uncategorized

The two video’s are very different, the first and best one is from the guy from the Science Asylum that is also a video channel. And for the first time I looked up what his name is and that seems to be Nick Lucid. The main reason for me to write this post is the fact that Nick is the very first person who tries to explain why electrons do anti-align. He thinks it has to do with the so called ‘intrinsic’ angular momentum that electrons have. I think that does not solve the energy problem in any serious shape or form, the energy problem is of course that the potential energy of the electrons gets raised in they turn into an anti alignment with the applied magnetic field.

To put this problem in a more simple to understand thing with say gravity: If you throw a piece of rock perfectly horizontal, it will move sideways and only down. If it goes up that would be a serious energy problem because they higher the rock is the higher it’s potential energy and where does the rock gets that energy from? Therefore in reality you never see rocks spontaneously fly up but in quantum mechanics with electrons and magnetic fields this happens all the time. Anyway this happens all the time if it was true that electrons are tiny bipolar magnets. And of course I don’t think that, I think that electrons carry a monopole magnetic charge just like they carry a permantent monopole electric charge. The only difference is that there are two magnetic charges that electrons can have where the electric charge is always the same and is negative.

I prepared 7 images but one was not needed so there only a few screen shots. So basically 6 pictures mostly text and a few basic calculations related to the supposed way electrons get accelerated by magnetic fields. There are an additional 3 Figures so all in all 10 pictures or images and two video’s.

The first video is the best, anyway in my opinion, that’s the one from the Science Asylum. I added the second video because that is one of those many video’s that simply skip where this all runs out of the rails: Why do electrons anti-align, where comes the energy from? So lets go:

Figure 1: Screen shots from the first video.
Figure 2: How to deal with the Lorentz force in a setup like this?
Figure 3: With this setup the Lorentz force is not a hinder.

The next image is just a leftover.

If you made it till here, you can now finally see the video from Nick. That is if the video would embed and for some strange reason it does say it won’t… Anyway the title of the video is Physics Misunderstood This Experiment For Years. (For the time being even the link does not work, it is now a private video… So may be next week I’ll give it another try and see if this is a temporary thing.

And the second and last video of this post. This is much more a demonstration of how the energy problems there are with this bipolar model for the electron are just never talked about. It is at the end of the video but there he does it.

Weirdly enough this video embeds seamlessly…

That was it for this first post of the year 2025. As always thanks for your attention.

Keep it simple: Take the line integral of the identity in two complex spaces.

3D complex numbers

This is now post number 278 on this website and to be honest the content of this post should have been here years ago. This post is about complex integration and I more or less compare how you do that in the complex plane against how it is done in 3D space.
Integrating the identity simple means integrating zdz on the complex plane and XdX on the space of 3D complex numbers. Of course I have used complex integration in higher dimensional spaces in the past when I needed it. For example this is how I found my first number tau: On the space of 3D complex numbers you must find the logarithm of j^2 (not j because j has a determinant of minus one) and I did that with complex integration.
For people who are new to this website: j denotes the imaginary unit in 3D space and it’s third power equals -1. That mimics the situation in the complex plane where the imaginary unit i if you square it that gives -1.

All these years I never used complex integration just to find a primitive, so that is done in this post. And since we are integrating zdz we expect to find 0.5z^2on the complex plane while on my beloved space of 3D complex numbers the integral of XdX should yield 0.5X^2.
Of course we will find these results because otherwise I would have been very stupid in say the last 15 years. Of course just like every body elso I have been very stupid on many occasions on such long timescales, but not when it comes to 3D complex numbers.

Oops, I see I still have to make the seven png pictures but that won’t take very much time. So that’s this math post: 7 pictures of each 550×1500 pixels in size. I hope that after reading it you can also perform complex integration is say the space of 4D complex numbers.
And now we talk about 4D numbers; I also included at the end the famous quaternions and of course if we try to integrate them we get the usual garbage once more demonstrating that when it comes to differentiation and integration the quaternions are just awful.

That was it for this math post. Likely the next post is another video where the famous Stern Gerlach experiment is explained. Of course in such videos they never jump to the correct conclusion that says it is very likely that electron magnetism is monopole in nature. Just like their electric charge by the way, of course for all professional physics persons the electron has to be a tiny magnet. Not that they have much so called ‘five sigma’ experimental evidence for that, but for them this is not a problem…

Ok, let me hit that button ‘publish website’ and may I thank you for your attention.

Integration and the number alpha.

3D complex numbers

It’s about time to write this post because the pictures were finished a few days back but I was a bit lazy in the meantime. In this post I only evaluate two line integrals both in some way related to the famous number alpha. And we do it only on the 3D space, there are much more numbers alpha on other spaces but we just do the complex and the circular multiplication in three dimensions.
In this post when it comes to the number alpha we mostly need the one property that alpha is it’s own square. Therefore you can break down all powers of alpha into alpha itself, this is very handy when for example you use a power series. As always X denotes a 3D number and one of the integrals we will look at is the exponential of alpha times X:
exp(alpha * X).
Why do we do this? Well try to find a primitive of the above exponential, that is a bit hard because another property of alpha is that it has no inverse and as such you can’t divide by alpha and so what to do?
The second example is integration of the most standard exponential exp(X) along the main axis of non-invertibles: All real multiples of the number alpha. For me this was a surprising result because it all becomes so much more simple. All in all this post is five pictures long but it is in the 550×1500 pixel size so they are relatively long. Ok that is all I had to say and let me now hang in the five pictures:

That was it for this post, thanks for your attention and may be see you in a future post.

Seven properties of the number alpha.

3D complex numbers, 7 Properties of Alpha

A long time ago around the time I started this website I had something known as “The seven properties of the number alpha”. But it was all spread out over two websites because until then I wrote the math just on the other website. But over the years I have advocated a few times to look up that stuff as the seven properties of alpha, so I decided to write a new post with the same title. I didn’t copy the old stuff but just made up a new version.
The post is amazingly long; a normal small picture has the size of 550×775 pixels, larger ones I use are up to 550×1100 but now it has grown to 11 pictures of size 550×1500. Likely this is the longest post I have written on math ever.
As always I have to leave a lot out, for example doing integration with the involvement of the number alpha is very interesting. During writing I also remembered that e to the power pi times i times alpha, wasn’t that minus one? Yes but we are not going to do six dimensional numbers or hybrids like the 3D circular numbers and replace the reals by the 2D complex numbers. Nope, in this post only properties from the 3D numbers alpha; the complex and the circular one.
More or less all the basic stuff is in this post; from simple things like alpha has no inverse to more complicated stuff like the relation with the Laplacian operator. I made a new category for this post, it even has the name “7 Properties of Alpha” so that in the future it will be even easier to find.

Lets hope it is worth of your time and lets hope you will find it interesting.

Figure 01: Alpha is the center of the complex exponential.
Figure 02: You can write X as the sum of two non-invertible numbers.

Ok, that was a long read. I want to congratulate you with not falling asleep. I have no plans for a next post although it is tempting to write something about integration along the line of real multiples of the number alpha.

Another video on the SG experiment and an additional pdf.

Magnetism

I didn’t plan on another ‘just a video post’ but I am still working on a new math post on the seven properties of the number alpha. All in all that is going to be one of the longest posts written, that why it takes a relatively long time. So that’s my excuse for another ‘just a video post’.

This is another video from Dr. Jorge S. Diaz and again the video is very good made with lots of interesting historical details. In it he shows what Gerlach thought before the experiment was done, that is in the first picture below. In case you did not know it: all that stuff with a varying magnetic field from strong to weak was thought out beforehand. Only the Stern Gerlach guys thought that the silver atoms themselves would act as tiny magnets. That’s why also in the simple math below you see the emphasis on the gradient of the magnetic field.
And that brings me once more to an important critisism, not on the video but on the lack of experimental proof that electrons are not accelerated in a constant magnetic field. Ok ok there is still the Lorentz force so that’s easier said then done but it is just missing. Just like there is no experimental proof or evidence that electrons are dipole magnets.

As all video’s on the Stern-Gerlach experiment this one too fails to explain as why tiny magnets would anti-align themselves with the applied magnetic field. After all this raises their potential energy and as such I consider this an important energy problem that you only have if you view electrons as tiny magnets.
I remember that back in 2015 when I myself did see this experiment for the first time, it was the fact that the silver atoms would more towards the weaker part of the magnetic field that I just could not understand. But in one or two days I had figured out that if you view electrons as magnetic monopoles, you don’t have weird problems. But back in the time I knew just nothing about electron spin so I had some learning to do.

In the video but also other sources say that it were Einstein himself and Heisenberg who pushed heave for a so called repeated or sequential SG experiment. On may occasions I have argued that such a repeated experiment has never been done in one century of time. And that is strange because if you were successful in this, you would almost one 100% sure have won a Noble prize. And that’s what all the physics people want; a Nobel prize… A repeated experiment would validate the probabilistic nature of electron spin, but it’s just not there and nobody in the physics community gives a shit about that. That’s why so often I say these people are talking out of their neck.

At about 19 minutes into the video Dr. Jorge S. Diaz claims that a repeated SG experiment has been done by Frisch and Segré but that is just plain wrong. In the Frisch-Segré experiment they tried to rotate the spin of an electron. But it failed so to bring this up as an example of a repeated SG experiment is not allowed.

That’s funny: Einstein & Heisenberg pushed for this…

Before I show you the video I want to make a quote from the pdf on the Frisch Segré experiment. The Frish and Segre guys claimed in an article to have observed nondiabetic spin flip. I doubt if that is true but at present day in a lot of physics labs they too think the can flip electron spin in say a diamond nitrogen vacency. But I think that when this happens, they have just another electron with the opposite magnetic charge.
This is a WordPress website and I believe they have a ‘quote environment’ hanging around somewhere. So lets try:

Immediately, Heisenberg and Einstein proposed multi-stage Stern–Gerlach experiments to explore deeper mysteries of directional quantization [2]. Ten years later, Phipps and Stern reported the first effort [8], which was unfortunately discontinued owing to Phipps’ involuntary return to the US [2]. A year later, Frisch and Segrè modified the same apparatus by adopting Einstein’s suggestion on the use of a single wire instead of three electromagnets to rotate spin; they also improved magnetic shielding, slit filtering, and signal detection [2]. Despite the use of three layers of magnetic shielding for the middle stage (i.e., the inner rotation chamber), the remnant or residual fringe magnetic field was still 0.42 × 10−4 T (or 0.42 G). Rather than fight the fringe magnetic field further, they took advantage of it. The magnetic field from the wire in the middle stage cancels the remnant field to produce a magnetic null point, around which the field is approximated as a magnetic quadrupole; consequently, they successfully observed nonadiabatic spin flip [9].

Well it is now high time for the video:

Multi-stage Stern–Gerlach Experiment ModeledDownload

In my mind or in my memory the Frisch Segré experiment failed so it is good to be corrected. But all in all you really can’t say that this was a repeated Stern Gerlach experiment. It was trying to flip the spin of an electron. And that while I think the magnetic properties of electrons are just like their electric properties: permanent and monopole.
Ok that was it for this ‘just a video post’. Thanks for your attention.

Video about the Stern Gerlach experiment, it’s good in the details.

Magnetism

One or two days back this video from Dr. Jorge S. Diaz came out and all in all in it’s kind it is very good. Even for me there is a lot of new stuff in although nothing of the real important things like why an inhomogeneous magnetic field was used: Otto Stern thought that the silver atoms themselves would act like tiny magnets because of the electrons going round the nucleus. I want to remark that using an inhomogeneous magnetic field when it comes to atom sized magnets makes sense, where I draw the line is the blind application to a point like particle like the electron.
So all the big hammers were already known to me yet there is a lot of cute stuff in it I had never seen. Things like the first introduction of those quantum numbers from the principle n to the magnetic number m.
In the video Jorge Diaz shows once more what the physics people use as the potential energy when a dipole magnet is placed in a magnetic field, you can see that in the picture below.

It is well known that nature loves to minimize the potential energy and here this is the case if both vectors mu and B point in the same direction. In that case the inproduct is a positive number and the minus sign guarantees the minimum of potential energy.
Last year I made a picture for repeated use during this year 2024 and in it you see the official version of an electron pair. The Pauli exclusion principle says that the magnetic numbers must differ and as such they must have opposite or anti-parallel spins. The whole problem is of course that if you calculate the potential energy where you view one spin in the magnetic field of the other and use the above expression, you get a positive potential energy. That’s weird since in the science of chemistry it is well known that the electron pair plays an important role in forming atomic and molecular bonds. Here is the sketch of the electron pair once more:

This potential eneregy isn’t minimized.

I made a similar picture for a lone electron that anti-aligns itself with an applied magnetic field:

Beside the potental energy problem, how can this be stable?

As you can see for yourself in the video below, people like Jorge Diaz never even mention that there are severe energy problems. I name that avoiding Crazyland, they only explain the things that sound logical and as soon as it becomes absurd like here with the electron pair, they just don’t talk about that.
The weird potential energy problems arise only if you think the electron is a tiny magnet. Since the year 2015 every year I became a bit more convinced that electrons are magnetic monopoles just like they are electric monopoles. All energy problems fade away fast if you do that but hey try to explain that to people like Jorge Diaz! Or for that matter all those other professional physics people out there, those weirdo’s also think that magnetic monopoles do not exist so the taks of explaining things to those people is an almost impossible task.
I also combined a few screen shots from the video with people that played some role or contributed to the Stern-Gerlach experiment. For myself I more or less like it that even a guy like Albert Einstein never realized the monopole nature of electron magnetism. But I am also well aware that this can work against me; the physics professionals will likely think that if Albert didn’t see it, it can’t be true and as such for themselves they have once more confirmed that magnetic monopoles don’t exist…

And finally the video, again in it’s kind it is a very good video:

A lot more could be said or written but lets not do that and may I thank you for your attention.

On the number of integer solutions on the ellipse x^2 + xy + y^2 = N.

2D multiplications

If you view the interger points of the plane equipped with the elliptic complex multiplication as prime numbers or composites, you have found a good basis to predict or calculate how much of such integer solutions there are. This problem is closely related to the number of integer points on a circle, I am sure you can find enough internet resources of that problem.
I decided to take a deeper look into this because of a video on the Eisenstein primes, those Eisenstein primes and integers look a lot like my own elliptic complex integers. But the Eisenstein numbers are defined on the standard complex plane while I modified the way the multiplication is done via a tiny change to i^2 = -1 + i. So that now ‘rules’ my plane with elliptic complex numbers and one of the nice properties is that i^3 = -1 while for that Eisenstein stuff you get +1. So the Eisenstein numbers are more a way of say the split complex numbers.
Lets start with the video that I considered to be remarkable good:

For me it was all brand new because to be honest I never ever studied the Gaussian integers and how that gives rise to a unique way of factorizing them. But now I had to solve the problem for my own 2D elliptic numbers I have to say that yes it is beautiful once you grasp what’s going on.

A well known fact from the Gaussian integers is that the number 5 is now no longer a prime number because it can be factored into 2 + i and 2 – i. On the complex elliptic numbers it is 7 that is the smallest integer to have that same fate as 5 on the Gaussian integers. This post is mostly about the two factors that factorize 7, I named them p and q and the interesting thing is they are each others conjugate.
But the number 3 is also no longer a prime so all in all for a person like me who dislikes algebra so often for so many years, it was all in all a nice patch of math to understand.

This post is 8 pictures long, each 550×1250 pixels and on top of that two extra figures are addes. So what more do you want? I kept the stuff as less technical as possible while hoping that if you want to dig a bit deeper you can do that for yourself now. So lets start the picture thing:

Figure 1: Seven is the smallest number that has two conjugate prime factors.
Figure 2: These are the 18 integer solutions to det(z) = 49.

Here is a link to the free Wolfram webpage where you can check that indeed 1729 has 48 integer solutions because you can make 8 different factors with the prime numbers involved: https://www.wolframalpha.com/input?i=x%5E2+%2B+xy+%2B+y%5E2+%3D+172

Here’s the wiki: https://en.wikipedia.org/wiki/Unique_factorization_domain.
That was it for this post, thanks for your attention.