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

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:

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.

A Cauchy integral representation for the 2D elliptic complex numbers.

This post is a bit deeper when it comes to the math side, I think you better understand it if you already know what such a Cauchy integral representation for the standard complex plane is. I remember a long long time ago when I myself did see this kind of representation for the first time, I was completely baffled by this. How can you come up with a crazy looking thing like this?
But if you look into the details it all makes sense and this representation is the basis for things like residu calculus that you can sometime use to crack an integral if all more easy approaches fail.
In most texts on the standard complex numbers (with standard I mean that the imaginary unit i behaves like i^2 = -1 whereas on my elliptical version the behavior is i^2 = -1 + i) it is first shown that you can take such integrals over arbitrary closed contours going counter clockwise. If the function you integrate has no poles on the interior of that contour, the integral is always zero.
I decided to skip all that although if you want, you can do that of course for yourself. I also skipped all standard proofs out there because I wanted to craft my own proof and therefore in this post we only integrate over the ellipses and nothing else.
Another thing to remark is that this is just a sketch of a proof, a more rigor approach would make the post only longer and longer and I think that people who are interested in math like this are perfectly capable of checking any details they think that are missing or swept under the carpet. For example I show in this post the important concept of ‘radial independence’ but I show that only for a very simple function g(z) = 1. It’s just a sketch and sometimes you have to fill in what is missing yourself. Sorry for being lazy but now already this post is 5 and a half pictures long so that’s long enough.

It also contains two extra figures and may be I will write a small appendix related to figure 1. But I haven’t done that yet so below is the stuff and I hope you like it.

Ok, all that is left is an appendix where I give a third parametrization of the elliptic complex exponential. It is just some leftover from some time ago when I wondered if the two coordinate functions might some some time lags of each other. And yes, they are. In the case of these elliptic complex numbers the time lag is one third of the period.

Before I end this post, why not place a link to all that official knowledge there is around the Cauchy integral representation there is. Here is a link:
https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula

That was it for this post, as always thanks for your attention.

Eigenvalue functions for the elliptic complex plane.

This is a very short post, in it I even joke this is done in “Tik Tok style”. It is about finding the eigenvalues for an arbitray elliptic complex number. Such numbers have matrix representations and as such they have also eigenvalues that live in the ordinary complex plane.
Here the elliptic plane is the same as we always studied the last couple of months, it is ruled by the imaginary unit i via i^2 = -1 + i. If you need the eigenvalues of such a number, instead of going through the calculation for eigenvalues every time, with the eigenvalue functions you just substitute it in and it spits out the two eigenvalues.
To be honest I did not explain in detail why it works, I hope it is rather obvious. Take for example two commuting (square) matrices A and B. They have the same eigen vectors (because they commute) and as such it is very easy to find the eigenvalues of any linear combination of A and B.

The post itself is only two pictures long and I included a third picture that I used on the other website. Beside a female robot the third picture contains another factorization of the equation of the ellipse that is the determinant of the matrix representations.
So this equation for the ellipse can be factored on the standard complex plane and also on the elliptic complex plane. The interesting detail is of course that on the elliptic complex plane you have integer coefficients while on the standard complex plane this is impossible.

Basically the eigenvalue funtions are both a map from the elliptic complex plane to the ordinary complex plane. If it was made by a professional math professor he or she would likely call it an isomorphism but I name them eigenvalue functions. In the past I also made them for the 3D complex and circular numbers and of course for the 4D complex numbers that were under study years ago.
Enough of the introdutionary talk, lets go:

Now I left a lot of stuff out otherwise it would not be a Tik Tok short kind of math post. But you can also use the elliptic complex plane as your primary source of eigenvalues. For example at the other side of our galaxy there lives an alien race known as the Orcs. And for some kind of religious reason these Orcs just don’t want to use circles because as they all know circles are evil. But they found the elliptic complex plane and they use that for solving eigenvalue problems like eigenvalues from square matrices or even stuff that we humans know as the Hamiltonian energy operator. That should work just as good as we humans do in using the complex plane we have over here where the complex exponential is a circle.
So let us now look at the third picture that has both factorizations in it:

Let me leave it with that and as always thanks for your attention.

The cousin of the transponent.

Likely in the year 1991 I had figured out that the conjugate of a 3D complex number could be found in the upper row of it’s matrix representation. As such the matrix representation of a conjugate 3D number was just the transpose of the original matrix representation. Just like we have for ordinary complex numbers from the complex plane. And this transpose detail also showed that if you take the conjugate twice you end where you started from. Math people would say if you do it twice, that is the identity operation.
But for the two 2D multiplications we have been looking at in the last couple of months, the method of taking the upper row as a conjugate did not work. I had to do a bit of rethinking and it was not that hard to find a better way of defining the conjugate that worked on all spaces under study since the year 1991. And that method is replace all imaginary units by their inverse.
As such we found the conjugate on 2D spaces like the elliptical and hyperbolic complex planes. And the product of a 2D complex number z with it’s conjugate nicely gives the determinant of the matrix representation. And if you look where this determinant equals one, that nicely gives the complex exponentials on these two spaces: an ellipse and a hyperbole.
Now when I was writing the last math post (that is two posts back because the previous post was about magnetism) I wondered what the matrix representation of the conjugate was on these two complex planes. It could not be the transpose because the conjugates were not the upper rows. And I was curious what it was, it it’s not the transpose what is it? It had to be something that if you do it twice, you do the identity operation…

All in all in this post the math is not very deep or complicated but you must know how te make the conjugate on say the elliptic complex plane. On this plane the imaginary unit i rules the multiplication by i ^2 = -1 + i. So you must be able to find the inverse of the imaginary unit i in order to craft the conjugate. On top of that you must be able to make a matrix representation of this particular conjugate. If you think you can do that or if you don’t do it yourself you will understand how it all works, this post will be an easy read for you.

It turns out that the matrices of the conjugate are not the transpose where you flip all entries of the matrix into the main diagonal. No, these matrix representation have all their entries mirrored in the center of the matrix or equivalently they have all their entries rotated by 180 degrees. That is the main result of this post.

So that’s why I named it the “Cousin of the transponent” although I have to admit that this is a lousy name just like the physics people have with naming the magnetic properties of the electron as “spin”. That’s just a stupid thing to do and that’s why we still don’t have quantum computers.

Enough intro talk done, the post is five pictures long and each picture is 550×1200 pixels. Have fun reading it.

That was it for this post, one more picture is left to see and that is how I showed it on the other website. Here it is:

Ok, this is really the end of this post. Thanks for your attention and may be see you in another post of this website upon complex numbers.

Another way of finding the direction of the number tau.

A bit like in the spirit of Sophus Lie lately I was thinking “Is there another way of finding those tangets at the number 1?”. To focus the mind, if you have an exponential circle or higher dimensional curve, the tangent at 1 is into the direction of the logarithm you want to find.
In the case of 2D and 3D numbers I always want to know the logarithm of imaginary units. A bit more advanced as what all started a long time ago: e^it = cos t + i sin t.
An important feature of those numbers tau that are the sought logs is that taking the conjugate always the negative returns. Just like the in the complex plane the conjugate of i is –i.

The idea is easy to understand: The proces of taking a conjugate of some number is also a linear transformation. These transformations have very simple matrices and there all you do is try to find the eigenvector that comes with eigenvalue -1.
The idea basically is that tau must like in the direction of that eigenvector.

That is what we are going to do in this post, I will give six examples of the matrices that represent the conjugation of a number. And we’ll look at the eigenvectors associated with eigenvalue -1.

At the end I give two examples for 4D numbers and on the one hand you see it is getten a bit more difficult over there. You can get multiple eigenvectors each having the eigenvalue -1. Here this is the case with the complex 4D numbers while their ‘split complex’ version or the circular 4D numbers have not.
Now all in all there are six examples in this post and each is a number set on it’s own. So you must understand them a little bit.
The 2D numbers we look at will be the standard complex plane we all know and love, the elliptic and hyperbolic variants from lately. After that the two main systems for 3D numbers, the complex and circular versions. At last the two 4D multiplications and how to take the conjugate on those spaces.

The post itself is seven pictures long and there are two additional pictures that proudly carry the names “Figure 1” and “Figure 2”. What more do you want? Ok, lets hang in the pictures:

Years ago it dawned on me that the numbers tau in higher dimensional spaces always come in linear combinations of pairs of imaginary units. That clearly emerged from all those calculations I made as say the 7D circular numbers. At the time I never had a simple thing to explain why it always had to be this pair stuff.
So that is one of the reasons to post this simple eigen vector problem: Now I have a very simple so called eigen value problem and if the dimensions grow the solution always come in pairs…

That was it for this post, likely the next post is upon so called ‘frustrated’ magnetism because the lady in the video explains the importance of understand energy when it comes to magnetism. After that may be a new math post on matrix representations of the actual conjugates, so that’s very different from this post that is about the matrices from the process of taking a conjugate…
As always thanks for your attention.

A look at the set of non invertible hyperbolic 2D complex numbers.

This is a horrible simple post, after all for the complex numbers ruled by i^2 = -1 + 3i all you have to do is look where the determinant of the matrix representaitons equals zero.
Well yes that is what we do, at present day there is that cute package or applet (in the past that was a so called computer program, why is all that kind of stuff an “applet” nowadays?) called DESMOS. With DESMOS the two lines that make up the set of non-invertibles is easy to graph.
These two lines are interesting because they are the asymptotes of all those hyperboles in this space, anyway those hyperboles that can be written as det(z) = constant. Furthermore the two lines where det(z) = 0 separate the parts of this complex plane where det(z) > 0 from the parts where det(z) < 0.

I took the opportunity to introduce a more comprehensive notation to denote such spaces of 2D complex and split complex numbers. I wrote it much more as a set like in set theory and it includes the ‘rule’ for the imaginary component.

For readers who are new to this website and don’t have a clue what ‘hyperbolic’ or ‘elliptical’ 2D complex numbers are, it is all basically rather simple:
These numbers are complex because i^2 = -1 + something, in the case of this post it is i^2 = -1 + 3i. These 2D complex numbers have matrix representations and the determinant of these matrices are constant along certain hyperboles. The case det(z) = 1 is very interesting for any of such a complex plane because that is a multiplicative group. Just like the unit circle in the standard or ordinary complex plane is a multiplicative group.

All in all this post has five pictures of size 550×1200 pixels and an additional two figures from graphs from the DESMOS applet.

For me it was funny to write some ‘high school math’ with just a few parabole kind of stuff in it. During the writing of this post I came across the idea of making a matrix representation of all that conjugating stuff, as such I found a beautiful but still extremely simple way to find the direction of the number tau in a particular space. The great thing is that this time it works in all dimensions so not only the 2D complex numbers but likely much much more.
Lets try to upload my post to the internet and may I thank you for your attention.

Integrals for the number tau for the 2D multiplication defined by i^2 = -1 + 3i.

Yes yes I know we already calculated the number tau for this space equipped with a hyperbolic multiplication. (That was a few posts back using matrix diagonalization.) But I had a few reasons to write this anyway, one reason was just curiosity. I wanted to know how those integrals looked and since we had calculated the number tau anyway we did not need to solve these integrals with pencil and paper.
I also wanted you to show how you can write the product of such a complex number z against it’s conjugate. On the standard complex plane this defines a circle and on our hyperbolic space it is of course a hyperbole.
At last I wanted to pen down the formula for finding the inverse on this particular hyperbolic complex number space. It looks just like the way this is done on the ordinary complex plane with the exception that if you calculate it the conjugate is a bit different.
For me it is funny that we have exactly the same looking formula for the calculation of inverses. All in all it shows that the fixation the professional math professors have on all that “The norm of the product is the product of the norms” kind of stuff is only true because on the standard complex plane the determinant equals the square of the norm of a complex number
z. In our present case of hyperbolic complex numbers we devide the conjugate by the determinant and those determinant define hyperboles and not circles. So nothing of that “The norm of the product is the product of the norms” kind of stuff. The deeper underlying mechanism is just always the determinant of the matrix representation.

It has to be remarked however that the study of normed spaces is important in itself and also in practice: If you can find a good norm for some difficult problem like the successive aproximations in say differential equations and you can prove using that norm the stuff converges, that is BINGO of course. Yet a norm is only a tool and not all there is inside that strange space known as human math.

The post itself is 6 images long, in it I have two (pairs of) integrals going from 1 to i. The integrals are of course the inverse of a complex number because the derivative of the logarithm is the inverse and we want to know the log of i because by definition that is the number tau.

I included a so called Figure 1 that show the evaluation of these integrals by the Wolfram package for definite integrals, it’s a handy online tool in case you don’t want to evaluate your integrals with pencil and paper.
The last image is from the other website where once more I want to insult the math professors just a tiny tiny bit by using the standard formula of finding the inverso on the standard complex plane.
So all in all this post is 8 pictures long.

That was it more or less for this post, I hope you are a bit more confident by now that you can actually integrate spaces like this more or less just like you do in the standard complex plane. The last image is from the other website, it contains a female robot. Why there are female robots is unknow to me, after all with other tools you never have females like have you ever heard of a female screw driver or a female mobile phone?

Ok, that was it for this post. Thanks for your attention and just like the female robot look up into the light and start thinking about the wisdom behind “The norm of the product is the product of the norms”.

Parametrization of the hyperbole where the multiplication is defined by i^2 = -1 + 3i.

With a new number tau (see previous post) there is always a log of an suitable imaginary unit found, but that does not mean you have a parametrization instantly. And with ‘suitable’ I mean the determinant of the imaginary unit must be one because I always want to make some complex exponential, in this case a complex exponential hyperbole.
The post is relatively short, I always try to write short posts and I always fail fail and fail in that. But now it’s only four pictures long so this comes close to a tiktok version of my ususal math postings. By the way it has nothing to do with this post, but do you like tiktok? I don’t like it very much, it is more for people that have a bit different mindset compared to the way my old brain works…
In this post I first neatly write down the parametrization using the sinh and cosh to express the two coordinate functions. And after that I more or less express it all as much as possible into the two eigenvalues of our beloved imaginary unit i that rules this plane via: i^2 = -1 + 3i.
May be you have never thought about imaginary units as having eigen values themselves. In that case I invite you to calculate the eigenvalues of the ordinary numbers z from the complex plane. You know that plane that is ruled by an imaginary unit i via the rule: i^2 = -1. You will find a very interesting answer and of course after that you wonder WTF are the eigen vectors?

But let us not digress and enjoy the beauty of a complex exponential that is a hyperbole in this case. Here we go:

I am sorry for those bad looking accolades, it is some small fault in the Latex math package or some other old computer feature. You see it in much more texts written in Latex; the { and the } are just not presented properly.
That was it for this post, as always thanks for your attention.

The number tau for the hyperbolic multiplication i^2 = -1 + 3i.

Some posts ago I showed you how you can calculate the number tau (always the logarithm of a suitable imaginary unit) using integrals for an elliptic multiplication. To be precise you can integrate the inverse of numbers along a path and that gives you the log. Just like on the real line if you start integrating in 1 and integrate 1/x you will get log(x). If you have read that post you know or remember those integrals look rather scary. And the method of using integrals is in it’s simplest on the 2D plane, in 3D real space those integrals are a lot harder to crack. And if the dimension is beyond 3 it gets worse and worse.
That is why many years ago I developed a method that would always work in all dimensions and that is using matrix diagonalization. If you want the log of an imaginary unit, you can diagonalize it’s matrix representation. And ok ok that too becomes a bit more cumbersome when the dimensions rise. I once calculated the number tau for seven dimensional circular numbers or if you want for 7D split complex numbers. As you might have observed for yourself: For a human such calculations are a pain in the ass because just the tiniest of mistakes lead to the wrong answer. It is just like multiplying two large numbers by hand with paper and pencil, one digit wrong and the whole thing is wrong.
Now we are going to calculate a log in a 2D space so wouldn’t it be handy if at least beforehand we know in what direction this log will go? After all a 2D real space is also known as a plane and in a plane we have vectors and stuff.

So for the very first tme after 12 years of not using it, I decided to include a very simple idea of a guy named Sophus Lie. When back in the year 2012 I decided to pick up my idea’s around higher complex numbers again of course I looked up if I could use anything from the past. And without doubt the math related to Sophus Lie was the most promising one because all other stuff was contaminated by those evil algebra people that at best use the square of an imaginary unit.
But I decided not to do it because yes indeed those Lie groups were smooth so it was related to differentiation but it also had weird stuff like the Lie bracket that I had no use for. Beside that in Lie groups and Lie algebra’s there are no Cauchy-Riemann equations. As such I just could not use it and I decided to go my own way.
Yet in this post I use a simple idea of Sophus Lie: If you differentiate the group at 1, that vector will point into the direction of the logarithm of the imaginary unit. It’s not a very deep math result but it is very helpful. Compare it to a screwdriver, a screwdriver is not a complicated machinery but it can be very useful in case you need to screw some screws…

Anyway for the mulitiplication in the complex plane ruled by
i^2 = -1 + 3i I used the method of matrix diagonalization to get the log of the imaginary unit i. So all in all it is very simple but I needed 8 pictures to pen it all down and also one extra picture know as Figure 1.

That was it for this post, we now have a number tau that is the logarithm of the imaginary unit i that rules the multiplication on this complex plane. The next post is about finding the parametrization for the hyperbole that has a determinant of 1 using this number tau.
As always thanks for your attention and see you in the next post.

Comparison of the conjugate on five different spaces.

To be a bit precise: I think two spaces are different if they have a different form of multiplication defined on them. Now everybody knows the conjugate, you have some complex number z = x + iy and the conjugate is given by z = x – iy. As such it is very simple to understand; real numbers stay the same under conjugation and if a complex numbers has an imaginary component, that gets flipped in the real axis.

But a long long time ago when I tried to find the conjugate for 3D complex numbers, this simple flip does not work. You only get advanced gibberish so I took a good deep look at it. And I found that the matrix representation of some complex z = x + iy number has an upper row that you can view as the conjugate. So I tried the upper row of my matrices for the 3D complex and circular numbers and voila instead of gibberish for the very first time I found what I at present day name the “Sphere-cone equation”.

I never gave it much thought anymore because it looked like problem solved and this works forever. But a couple of months ago when I discovered those elliptic and hyperbolic versions of 2D numbers, my solution of taking the upper row does not work. It does not work in the sense it produces gibberish so once more I had to find out why I was so utterly stupid one more time. At first I wanted to explain it via exponential curves or as we have them for 2D and 3D complex numbers: a circle that is the complex exponential. And of course what you want in you have some parametrization of that circle, taking the conjugate makes stuff run back in time. Take for example e^it in the standard complex plane where the multiplication is ruled by i^2 = -1. Of course you want the conjugate of
e^it to be e^-it or time running backwards.

But after that it dawned on me there is a more simple explanation that at the same time covers the explanation with complex exponentials (or exponential circles as I name them in low dimensions n = 2, 3). And that more simple thing is that taking the conjugate of any imaginary unit always gives you the inverse of that imaginary unit.

And finding the inverse of imaginary units in low dimensions like 2D or 3D complex numbers is very simple. An important reason as why I look into those elliptic complex 2D numbers lately is the cute fact that if you use the multiplication rule i^2 = -1 + i, in that case the third power is minus one: i^3 = -1. And you do not have to be a genius to find out that the inverse of this imaginary unit i is given by -i^2 .
If you use the idea of the conjugate is the inverse of imaginary units on those elliptic and hyperbolic version of the complex plane, if you multiply z against it’s conjugate you always get the determinant of the matrix representation.
For me this is a small but significant win over the professional math professors who like a broken vinyl record keep on barking out: “The norm of the product is the product of the norms”. Well no no overpaid weirdo’s, it’s always determinants. And because the determinant on the oridinary complex plane is given as x^2 + y^2, that is why the math professors bark their product norm song out for so long.

Anyway because I found this easy way of explaining I was able to cram in five different spaces in just seven images. Now for me it is very easy to jump in my mind from one space to the other but if you are a victim of the evil math professors you only know about the complex plane and may be some quaternion stuff but for the rest you mind is empty. That could cause you having a bit of trouble of jumping between spaces yourself because say 3D circular numbers are not something on the forefront of your brain tissue, in that case only look at what you understand and build upon that.

All that’s left for me to do is to hang in the seven images that make up the math kernel of this post. I made them a tiny bit higher this time, the sizes are 550×1250. A graph of the hyperbolic version of the complex exponential can be found at the seventh image. Have fun reading it and let me hope that you, just like me, have learned a bit from this conjugate stuff.
The picture text already starts wrong: It’s five spaces, not four…

At last I want to remark that the 2D hyperbolic complex numbers are beautiful to see. But why should that be a complex exponential while the split complex numbers from the overpaid math professors does not have a complex exponential?
Well that is because the determinant of the imaginary unit must be +1 and not -1 like we have for those split complex numbers from the overpaid math professors. Lets leave it with that and may I thank you for your attention if you are still awake by now.