I've prepared a DVD ISO of free ebooks, and also share my GitHub project

In my previous blog posts I've shared interesting fractal images, and now I'm sharing the JavaScript code (or at least, part of it) that has generated them (but I've made some modifications):

https://arpadfekete.github.io/ASMA/

I also have JavaScript code for using a different matematical idea to generate interesting pictures, but it is not clean enough for sharing on GitHub immediately, and I'm quite busy in these times with more important tasks, so I don't know yet when will I upload it.

I'm planning to read ebooks or listen to audiobooks or maybe do something for the environment or enter literary competitions or translate my own ebooks in my free time these days, except on Saturdays, when rest and/or do Math (mainly offline or at least not looking at the screen, but listening to podcasts). Among the ebooks, I would like to focus on the free ebooks made by the Project Gutenberg, and I've already pre-selected a lot of ebooks for me to read. I've even written a DVD of them, the ISO file of which I'm also sharing here:

https://drive.google.com/file/d/1bNGIhiFQ6PTg82zElENP9-TxVwlLaZDz/view?usp=sharing

This way I hope that I promote the wisdom of humankind, so this collection of non-fiction ebooks gets more attention. Note that these ebooks are shared by the terms of Project Gutenberg, so you should not ask money for them.

The relation between escape time fractals and chaos game fractals

I had the insight that both escape time fractals and "chaos game" fractals involve some kind of iteration. Some fractals can even be visualized by both ways: the escape time algorithm and the chaos game. One example of this is the Sierpinski triangle, which (or its approximation) I've visualized by the chaos game in a previous blog post, as well as by the escape-time fractal algorithm of the fractal zoomer software called XaoS (although the latter actually showed the areas that are NOT part of the fractal, with different colors depending of the escape time). Yesterday I had the idea what if I tried to visualize the Mandelbrot set by the chaos game, especially the points that are part of that fractal by different colors (so points that are usually black in other fractal generators, as those software visualize those points that are not part of the fractal instead, by different colors).

I had the insight that we could continue the iteration of the escape-time algorithm of the Mandelbrot set in the opposite direction! So, instead of taking:

z=z^2+c

we could take:

z=sqrt(z-c)

Okay, but there are two square roots of a complex number! Well, it is just handy for us, because we can introduce a random element in the iteration that is required by the chaos game anyway. So we take one of the square roots randomly. Thus I tried to implement it by JavaScript yesterday, but always the same picture appeared, no matter what was the initial value of z and what was the constant c:

Soon I realized that I've forgotten to subtract c in the algorithm, so different pictures appeared afterwards, depending on the random c value (but note that they did not much depend on the random initial value of z):

Still, the images did not look nice enough, as you can see, because there were too little amount of points belonging to the resulting fractals. So I had the idea what if I take:

z=cbrt(z-c)

Where cbrt is the cube root of its argument (I just took the complex cube root formula from WikiPedia, just like the complex square root formula). Again, there are three cube roots of a complex number, and it is good for us. This was handy for two reasons: it produced more points, and it made it easier to color the points, as there are 3 main colors: red, green and blue, just like there are 3 possible ways of taking the cube root. Now this method produced nicer pictures than the previous one:

Notice that the second picture of this kind is zoomed. I've implemented basic zooming, but I emphasize that the chaos game is generally poor in zooming, because the random iteration always wanders in the full area of the fractal, so we cannot zoom deep. That's why my method will probably not be a very popular one, and it will not replace the escape time fractal algorithm.

By the way, these pictures look like the Julia sets of the fractal z=z^3+c.

The previous pictures are of 640x480 resolution, but I've made two pictures for those who like changing wallpapers. So:

 

 

That's about the relationship of the escape time fractal iteration and the chaos game fractal iteration.

More playing with the Chaos Game

This blog post is a follow-up for two previous blog posts: "I was programming fractals a bit" and "More fractal wallpapers related to the Sierpinski Triangle". First it came to my mind that I could transform the coordinates of the next iteration point by taking the distance between 0 and 1, then executing "y=Math.sqrt(x*(2-x))" or "y=1-Math.sqrt(1-x*x)" on the distance, and then getting the transformed point based on the new distance. However, this idea did not provide nice pictures, unless I made a mistake in the algorithm:

1. Fractal-like picture by mistake:

2. Fractal-like picture by mistake:

After I've figured the mistake out, I realized that I can make nicer fractals if I get a weighted arithmetic mean between x and y as described above (because my original transformation was too much). So here are some nicer pictures:

3. (Math.sqrt(num*(2-num)) + 4*num) / 5:

4. ((1 - Math.sqrt(1-num*num)) + num) / 2:

5.  The same as above, but added triangles in the centers:

So these are the ideas I had for triangles, not as nice as expected. Another idea has come to my mind that I could also do the same with regular heptagons and the colors of the rainbow instead of just red, green and blue in the corners. I did not feel like to spend the time computing every ratios exactly, so the following pictures are the result of experimentation with shrinking values:

1. Regular heptagon fractal wallpaper on black background:

 

2. Regular heptagon fractal wallpaper on white background:

I think that's all for today... but I also have other ideas for programming math pictures, not only fractals... I may implement them if I have free time for it, and feel like to do it.

I didn't win on the Gravity Research Foundation 2020 Awards for Essays on Gravitation

I've found this essay among my old emails (in PDF), and I thought why not publish it here?

On the gravitational effects of third bodies on two specified bodies in space, and their consequences

Abstract:

In a system of many objects or bodies, the movements of two bodies close to each other are determined not only by their own masses, locations and velocities, but also by the masses, locations and velocities of all the other (third) bodies in the system. In some cases the gravitational fields of these third bodies act to increase attraction between the two specified bodies, but in some other cases they act as if to cause some degree of repulsion between the two bodies, like a tidal force. The author argues that this might be the real cause of many flat structures in the Universe, like the rings of Saturn, the planets of the Solar System, or the stars of the Milky Way.

Content:

Gravity is not as simple as calculations with the center of mass. Even in case of three bodies, the resulting problem of calculating their motions turned out to be very difficult, called the three-body-problem [1]. For the same reason, and also because I am not a Physicist (etc), I cannot come up with final calculations on the subject in the meantime. I just have one simplified example as follows: Let A and B be two bodies (actually point masses) of mass m, and their distance be 2s. Let O be the midpoint of the AB segment. Let c be a circle with midpoint O and radius 3s. For the sake of simplicity, our third bodies will be on the circle c. If the circle had a larger radius, our results would be similar, just different in scale. Let C1 and C2 be the intersections of the line AB and the circle c. Let e be a line through O perpendicular to the line AB, and let C3 and C4 be the intersections of the line e and the circle c. Let C1, C2, C3, and C4 also be point masses, or third bodies of mass m, and of course, let the initial velocities of A, B, C1, C2, C3 and C4 all be zero. Here is the figure:

Now we could apply the Newtonian equation of universal gravitation [3] to (approximately) get the forces which would move our bodies or point masses (F=G*m^2/r^2). C1 attracts A by F_A=G*m^2/(2s)^2, while C1 attracts B by only F_B= G*m^2/(4s)^2. As a result, C1 will cause A and B to increase their distance (of course, this will not be realized as A and B also attract each other, but still, their distance will be greater with C1 than without it). C2 will behave the same way as C1 did. However, C3 and C4 are in another position, and they will make A and B to be closer to each other than without C3 and C4. As for other possible bodies, on the circle c or farther, their effect on A and B will be somewhere in-between the repulsive (tidal [2]) effect of C1 and C2, and the attractive effect of C3 and C4. What does this mean? To me, it probably means that three-dimensional structures (with alike sizes in each dimension) tend to collapse, but flat structures tend to be preserved more. This could be the cause of why many structures in the Universe are (somewhat) flat: the Milky Way, the Solar System, and the rings of Saturn. Of course, this is harder to see in three dimensions with all the bodies having initial velocities (in different directions) and different masses, so I may let the evaluation of this to my Physicist readers. (Note that the WikiPedia article on „tidal force” [2] also mentions that the ring system of Saturn is caused by the tidal force, however, another WikiPedia article says that there is still no consensus as to how the rings of Saturn were formed [7].) Why is this important? Because otherwise the flatness of the celestial structures may be explained by other theories, like that which I have watched in a YouTube video by „minutephysics” entitled „Why is the Solar System Flat?” [4]. It says that every three-dimensional cloud of matter has a plane which defines its total angular momentum, which is preserved due to the law of conservation of angular momentum [5]… that's OK... but it also says that the matter not on this plane gets closer to it due to collisions between celestial bodies (maybe coming from different sides of this plane). Here I would argue that no collisions are necessary, but celestial bodies might arrange themselves near a plane for reasons I described above, i. e. the gravity of third bodies (but afterwards, they might collide as well, of course, especially when they enter into the stronger gravitational field of one another). Also, the „collision theory” might not adequately explain why the rings of Saturn are composed of so small objects, or why the asteroid belt [6] is there between the orbits of Mars and Jupiter. Well, I have come up with this idea of mine in 2014 or 2015, and included it in my math blog, and I might also have shared it on UseNet. Since then, I deleted that math blog of mine, and it was offline for about 5 years. Recently I have put some parts of it on the net again, with a different address, and some edition, so you could read my brief mentioning of this topic there [8]. I am not a Physicist, so I think it would be good if this topic were reviewed by some Physicists competent on this area.

References:
[1] Three-body problem, WikiPedia
https://en.wikipedia.org/wiki/Three-body_problem
[2] Tidal force, WikiPedia
https://en.wikipedia.org/wiki/Tidal_force
[3] Newton's law of universal gravitation, WikiPedia
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
[4] „Why is the Solar System Flat?” by „minutephysics”, YouTube
https://www.youtube.com/watch?v=tmNXKqeUtJM
[5] Conservation of angular momentum, WikiPedia
https://en.wikipedia.org/wiki/Angular_momentum#Conservation_of_angular_momentum
[6] Asteroid belt, WikiPedia
https://en.wikipedia.org/wiki/Asteroid_belt
[7] Rings of Saturn, WikiPedia
https://en.wikipedia.org/wiki/Rings_of_Saturn
[8] Noble challenge: search for an error in Mathematics (my blog post mentioning this idea) from the blog „Recreation in Mathematics by Árpád Fekete” (dead link removed)

More fractal wallpapers related to the Sierpinski Triangle

You may have seen the result of my playing with the Chaos Game, to make some fractal wallpapers related to the Sierpinski Triangle.

Today I've felt like experimenting a bit more with this, using some newer ideas. I'm going to share the result here, 4 new fractal wallpaper pictures. (The JavaScript code is not looking very nice, but if I'll feel like to clean it, I may put it on GitHub in the future...)

1. Filling the center of the Sierpinski Triangle as well, but change the ratio of shrinking so that there should be black areas in the fractal as well.

2. Partially filling the center of the Sierpinski Triangle.

 3. Rotate the smaller triangles in the Sierpinski Triangle.

4. Use different shrinking ratios, so this looks like like cell division / fission.

Summary: I could not make more interesting images than in my previous blog post about this (linked in the beginning), but at least I played a bit and used some Mathematics (without having proof about it in my mind).

I did not find interesting polyhedra containing the regular heptagon

As I was thinking more about this, what I also mentioned in an earlier blog post of mine (about polyhedra), I realized that I cannot find interesting new polyhedra that are not yet discovered, and somewhat regular, and maybe contain the regular heptagon. For example, I was thinking about three different regular heptagons ABCDEFG, HIJKLMN, OPQRSTU in 3 dimensions where A=H, B=I, O=K, P=J, Q=C, R=D. Then I conjectured that this way of putting together regular heptagons (so that at each edge, two-two heptagons should meet this way, by this angle) could be continued (in a way that these heptagons can intersect), but would end after some time, and the result (all of the points and the heptagons) would make an interesting (probably concave) new polyhedron. Whether or not the process ends, I planned to check by computer (approximately).

However, today I approached the problem differently. I asked the question: on the plane, can regular heptagons be put together at their edges, so that two-two regular heptagons should meet at each edge, and in a finite part of the plane there should be a finite number of heptagons? Then I realized that there will be 14 edges around a vertex of a heptagon, and the angles between these edges will be π/7. The angles of the heptagon are (2*π)*5/14, or π*(7-2)/7, by the way (and 5 and 14 are relatively primes). Then I found that two adjacent edges (around a vertex) will form a triangle whose angles are (π/7, 3π/7, 3π/7) and therefore, the third edge of this triangle will be shorter than the edges of the heptagons. However, these shorter segments are also around a vertex of the heptagons, connecting other vertices of the heptagons to it, and there are again 14 of them, with angles π/7 between them. We can do this process again and again, getting smaller and smaller segments. Therefore, there should be an infinite number of vertices, if we try to put together heptagons on a part of the plane (even if they intersect).

Well, it has come to my mind that I could try to reason somewhat like this (by analogy), about heptagons in the 3D space (or on a spherical surface). Of course, this problem is harder, so I shall suppose that the edges of the heptagons are the smallest distances between two points of the supposed resulting polyhedron. Otherwise I may find smaller and smaller segments by a way analogous to the previous reasoning, and conclude that the process of forming the polyhedron will not end. Then I realized that it is not possible for 3 or 4 regular heptagons to meet at one vertex. It is possible for 5 or more heptagons to meet at one vertex, if these heptagons intersect, and also intersect with the diagonals of the pentagon (or more) that is projected around this vertex. However, 6 or more heptagons meeting at one vertex will make smaller segments (as distances between vertices) than the edges (sides) of the heptagons. The only possible way seemed to be 5 heptagons meeting at one vertex, but I checked this possiblity and it looked impossible. Of course, all of this reasoning of mine is not Mathematically precise or perfect, but it is enough for me to reject and cancel further work on this topic, as my time is very precious.

What's next for me in Mathematics? I think I'll train myself in Number Theory (both by reading books about it and by solving KÖMAL exercises of type B, or later even of type A). (Fortunately, I have a lot of books and ebooks about Mathematics now, I do not need more for a long time.) Then I'll probably try to prove or disprove the Beal conjecture or the Goldbach conjecture... but if Mathematics remains hard for me, maybe I'll try to get rich quick by other means, too (e.g. writing books or composing music by LMMS).

I didn't win on the Alpine Fellowship Writing Prize, 2025 (On Fear)

I didn't win on the Alpine Fellowship Writing Prize, 2025 (on the topic "Fear"), I got a notification about it today. Thus I'm going to share my essay here, because it is non-fiction (but I do not plan to share my losing poetry or fiction):

VIPs can fear

I fear of dogs, especially the large ones like a pit bull, especially when there are more of them. I would like to show you that I am right. Flies do not seem to fear, neither do locusts. That is because it is a good strategy for them not to fear in the struggle for existence. Many other animals fear, however, humans included. That is because evolution favours fear in many cases. Thus it is reasonable to fear sometimes, and it is reasonable not to fear at other times. Still, there are people who tend to fear more often, they are the cowards. Traditionally, cowardice is considered a bad trait, while its opposite, courage is considered a virtue since at least Aristotle. Indeed, Aristotle taught courage (and not audacity), and his disciple, Alexander the Great successfully applied it to spread the ancient Greek language and culture more in antiquity. However, this fact does not justify the claim that courage is better than cowardice in general, or in our case of fear of dogs. We need to examine the question further.

As far as we believe, Socrates, Aristotle's teacher's teacher, said that "Virtue is knowledge", and Aristotle himself taught that "Courage is a virtue", so if both Socrates and Aristotle were right, then we could infer that "Courage is knowledge". According to this theory, the people who know more about dogs and about fighting may be usually less afraid of dogs than those who only have their instincts of fear. For instance, news about someone strangling a cougar has given me some more courage against dogs. Other examples may be found in the novel "Call of the Wild" by Jack London, so this could also help, at least when we have clubs. (People with clubs could easily beat dogs in that novel.) Still, there are many times when I fear of dogs and I don't think that more knowledge could help, except for knowledge about Heaven. There is also knowledge that increases fear, like the Wikipedia List of fatal dog attacks including a lot of cases about pit bulls. Simply, there may be cases where the chances of dogs are better in a fight. Consequently, cowardice in these cases may be a virtue, if virtue is still knowledge.

Perhaps it will be enlightening to you if I share the details of some specific cases where I fear of dogs. The shortest route from/to the city center passes by a house with a German Shepherd dog there. The gate is usually closed, but sometimes it was open and the owner of the dog was there. Therefore, I usually take another route. (Once I went that way in spite of the dog and the gate open, because I was in a depressed mood.) In other cases, we went to my grandmother's house by bike. There is a house with many big dogs on the way, where the gate is usually closed, but sometimes it was open and the owner of the dogs was far away in the house. Therefore, I prefer not going that way, if I can... and I do not bless those who keep big dogs.

Am I a coward? This really seems to be so, because I am also timid when approaching women. But I can explain. One of my acquaintances has told me that he drank alcohol to pluck up courage to talk to a girl. Our fear of courtship might be justified when we unconsciously feel that the girl or woman is not the right companion for us. Otherwise, our Nature may allow us to overcome this fear anyway. Now if not everyone can be our partner, then we are VIPs (very important persons), at least to ourselves! Thus the lack of bravery might mean something positive, something of value! VIPs also deserve more protection from dogs and from other attackers. Fear can mean such a protection. VIPs should not depend on the mercy of dogs!

Once I walked in the city, and I saw a German Shepherd dog (or a similar one) on the other side of the street closed in a large yard. As far as I remember, it saw me, too. I raised my arm and shaped my fingers as if I were holding a sword. I imagined that it was the Sword of God (Sword of Attila) or a lightsaber (from Star Wars). Then the dog seemed to fear and whine.

I was programming fractals a bit

It is easy to draw a Sierpinski triangle by programming, there is an easy algorithm for that called the Chaos game. By enhancing this method, I generated some fractals today afternoon. My main goal was to prepare a picture for a physical present that I'm going to give to someone. However, since I've written the program, I've also generated some wallpapers for myself. I'm also going to share these pictures on this blog. First comes the original Sierpinski triangle, colored by my idea:

Next comes the "Golden Sierpinski triangle" that I also added earlier to the fractal drawing software XaoS (but XaoS uses a different algorithm and different coloring). It uses the reciprocal of the golden ratio for shrinking instead of 1/2:

Next comes the same fractal, if I do not run the Chaos game long enough:

 

Happy PI Day, 2025!

Many ways to practice Mathematics (some interesting links)

I thought that I knew all the best English-language and German-language podcasts about Mathematics, but some days ago I've found two that are not only alternatives, but may also be better than the others! The first one is called "My Favorite Theorem", and it is accessible on the following link:

https://kpknudson.com/

The other is called "Breaking Math", and it is found here (Maybe I knew the title of this one, but I could not access it on another location for dead links):

https://www.breakingmath.io/

Maybe there will be more. Thus I'm going to listen to these podcasts (EDIT: except Breaking Math, as it is only accessible through service providers which prohibit use for commercial purposes), even if I did not plan to listen to new podcasts in 2025. Earlier I have also started to read free google books, and I first read about the history of Mathematics. As far as I remember, I read "A General History of Mathematics" by John Bossut in 2024, and I'm currently reading "A Short Account of the History of Mathematics" by W. W. Rouse Ball, and I like the latter more than the former, so I can recommend the latter (maybe it is best to read both). I've also read "A History of the Mathematical Theory of Probability" by I. Todhunter in 2025, but although it has broadened my horizons, it was not entirely intelligible to me, due to its complicated integrals and other things hard to understand. It did not make me feel like doing this kind of Mathematics, although I still had a takeaway: the St. Petersburg Paradox. Apart from these resources, I've also found a very good resource for Mathematics, which can be found here:

https://planetmath.org/

It has a similar licence as WikiPedia, so maybe these two are the best places to read about advanced Math. I especially liked the proof of Lagrange's Four Square Theorem here:

https://planetmath.org/proofoflagrangesfoursquaretheorem

Recently I was also interested in Symmetry Groups and Point groups in three dimensions, but it is still hard to understand for me, whether these theories prevent me from discovering such polyhedra that are highly symmetric, but yet unknown. After these articles, it seems to be harder to imagine that I will be able to discover anything new in this field. That's why I'm also turning part of my attention to Number Theory, first by solving easier exercises, e.g. from KÖMAL. By the way, I have also received an old Number Theory textbook for Christmas (translated to Hungarian from Russian).

By the way, after listening an episode of the "My Favorite Theorem" podcast about the Gauss-Bonnet Theorem, I have found (again) that my discovery that I mentioned earlier on this blog is already known as Descartes's Theorem on the "total defect" of a polyhedron. So this was a summary of my recent Math life.

About Writing, I have also some things to share. Nowadays there are less international essay contests for adults (with acceptable terms) than there were earlier, but maybe it's good to know that there may also be opportunities to publish works in journals, and the best articles in those journals may win prizes. I've found the "Royal Economic Society Prize" and the prizes of "The American Finance Association". Apart from these, I could also subscribe to the newsletter of the "Independent Social Research Foundation", maybe there will also be some good opportunity there.

The Brahmagupta–Fibonacci identity and "Binary quadratic forms"

I really like Fermat's theorem on sums of two squares... and its proof probably requires the Brahmagupta–Fibonacci identity (or by another name, Diophantus identity), so I like that as well. When looking at this identity and its generalizations, it has come to my mind that I may be able to generalize it as well, or at least discover something similar. The Brahmagupta-Fibonacci identity says that the product of two integers that are each expressible as the sums of two squares, so this product is also expressible as the sums of two squares. This can be checked by practicing basic algebra:

(a²+b²)(c²+d²) = (ac-bd)²+(ad+bc)² = (ac+bd)²+(ad-bc)²

Brahmagupta actually knew a more general identity:

(a²+nb²)(c²+nd²) = (ac-nbd)²+n(ad+bc)² = (ac+nbd)²+n(ad-bc)²

So I made the conjecture that the product of two integers that are each expressible as (x²+xy+y²), for different integer values of x and y, so this product is also expressible as (x²+xy+y²) for some integer values of x and y. My conjecture was justified by checking it for small values of x and y. But it was harder to prove the conjecture, as I had no formula to check. However, by looking at many examples, I managed to guess the formula, so I could check it then:

(a²+ab+b²)(c²+cd+d²) = (bd-ac)²+(bd-ac)(ad+bc+ac)+(ad+bc+ac)² = e²+ef+f²

where:

e = bd-ac and f =ad+bc+ac

Then I was thinking about how useful this result might be... and I have also checked WikiPedia more, maybe I find something about it... and I've found! Actually, Gauss and Lagrange were working on something even more general, i.e. (integral) binary quadratic forms, so they probably also knew what I "discovered" here. Still, it was good to take care of it, as this feels better than solving crossword puzzles, for example.

By the way, I also checked numbers of the form x²-xy+y², and I have found that:

• They are always positive or zero, as when xy is positive, then either x²>xy or y²>xy
• They are exactly the same numbers as those of the form u²+uw+w², because we can convert them this way: x²-xy+y² = (-x)²+(-x)y+y² = u²+uw+w² where u=(-x) and w=y. Still, it is interesting that they also seem to be the same numbers for only non-negative values of x, y, u and w.

What do I do in Mathematics in these times?

Although today I had other things to do (visit my relatives), on most Saturdays I'm planning to do some Math and try to do it offline without computers, if possible. Thus I would observe the "Sabbath" this way (ideally, I should not turn on my laptop on Saturdays, and Mathematics can be an ideal activity I could do on those days). What these Math activities can be?

• I've downloaded a lot of lecture notes and books about Mathematics from the Internet (mostly in English), but only those which have a good license like CC-BY, CC-BY-SA, GNU FDL, public domain, Project Gutenberg license or downloaded from "free Google books" (which I assume should be almost public domain). These files occupy about half the size of a DVD on my disk (and more, since I have more copies of them). Thus I'm occasionally also reading some of these books, or their printed versions, which I received as presents. (So they asked me what should they give me as a present and I wrote that they could print some of these books for me like university students print their lecture notes.) Earlier I also printed some of these myself, but those are just separate pages in folders.
  

• I've also picked some good-looking Math exercies from the website of KÖMAL (I linked the English-language version for you, but I picked them in Hungarian, of course) (EDITED 2024-12-07). Thus I can think on them on Saturdays. Many people regularly solve crossword puzzles, Sudoku and the like, but in my opinion these KÖMAL exercises may be much more interesting and challenging than those. As a compensation, I'm planning to order 1% of my taxes to the foundation that runs KÖMAL (MatFund) in the next year (in Hungary, we can order 1% of our taxes to a NGO we choose, and another 1% to a Church we choose). In this year, however, I already ordered this 1% to an environmentalist organization (so 1% of my taxes for the year 2023 is given to Environmentalists, but 1% of my taxes for the year 2024 will probably be given to Mathematicians in 2025).
  

• Sometimes I am also thinking on really hard Math problems, like the Goldbach conjecture (by the way, I prefer those which can be formulated by relatively simple statements, so I'll probably not be thinking on the Riemann Hypothesis in the near future, because it is not "elementary"). Lately I had not many ideas for them, so they were neglected... also because I realized that the KÖMAL exercises are still hard for me, so it is not likely that I can solve these much harder problems in the meantime. (But I acknowledge that even if I cannot solve the Goldbach conjecture, I can become wiser if I try it and learn new things along the way.)
  

• ... and the following is what I am writing this blog post for: I've begun to think about new types of polyhedra, are they possible or not? This is what I could try to research. Thus I have some ideas as seeds (top secret), and I could try to do Math to decide if these ideas lead to new types of polyhedra or they fail. During this research I've found the Virtual Polyhedra website of George Hart, which is very interesting and well done! It lists a lot of polyhedra types I previously knew nothing about (although I knew the Archimedean solids and their duals, for example), and the interactive visualizations are outstanding. However, my ideas are not there, so they will either fail or they are still very well concealed by God. Mathematicians teach that the symmetry types of possible polyhedra may not be very different from what we already know about, but I would still like to check this myself. However, I may not be skilled enough in Mathematics to do it, so it is possible that I will use computers to help me... but then I should probably not do this on Saturdays. On other days, I still focus on writing a new book about the critique of Christianity, so it is possible that this polyhedra goal of mine will progress very slowly, and it will still be an open question for me in the next year... but at least I wrote about this today so that my readers know that I may be about to discover something, which may be more worthwhile for me than thinking on the Goldbach conjecture.
  

By the way, I could also do some other random things related to Mathematics not listed here. For example, I recently realized that I couldn't have beaten the best 8th graders in Mathematics on the Zrínyi 2021 competition for elementary schools (so I am probably weaker in this type of Mathematics than I was in elementary school). I've also recently mentioned some of my activities or plans related to Mathematics on my Hungarian blog (including some logic puzzles on computer or physically). Since then, I've also found a new game called "Chroma" in Ubuntu Linux, that is also looking nicely (if I had time for it)! On my Hungarian blog, playing chess is also mentioned, which I try to do regularly on LiChess (usually about one game per day). It would be good to level up my chess skills to be able to win a competition in my town, but I will probably not go to local chess competitions until I play good enough on LiChess or against a computer on Normal level.

By the way, knowledge in Mathematics is something that is good to have. Even if this World is evil, Mathematics is good, so we can build up knowledge about Mathematics in our brains, and this way we can get close to the universal values that are true in all possible worlds, not just ours. Thus I've also begun to read a book about Mathematics that contains many things I already know, but it was still good to read it, because it was organized in a special way. (E.g. it has chapters about the different proof techniques, which is a good summary of them.) Thus Mathematics is beautiful.

The Mandelbrot VS the Triceratops fractal in XaoS: what makes the difference?

Many years ago I played with the source code of the XaoS fractal zooming program... I searched for new (fractal) formulae, and viewed how they look like as fractal images. I had the idea that instead of f(z)=z²+c (the iteration formula for the Mandelbrot set), I could use the generalization of triangular numbers (for complex numbers), so I used triangular numbers instead of squares. The formula for triangular numbers is n*(n+1)/2, so the iteration formula for my new fractal has become f(z)=(z²+z)/2+c. A very interesting fractal formed, which I called Triceratops. This was interesting because it was very different from the Mandelbrot set...

The Mandelbrot set

The Triceratops fractal

Of course, we need to zoom in to see the interesting parts of the fractal, like:

These are kinds of images we do not usually see in the Mandelbrot set. Later I realized that this is because the escape condition was too low for the Triceratops fractal... the escape condition for the Mandelbrot set is 4, which means that the iteration stops when |z|²>4, or abs(z)^2>4. In the case of the Mandelbrot set, this is equivalent to saying that the point is not part of the fractal, because it diverges to (some) infinity... and therefore, it is colored by some color different from black (depending on the number of iterations needed to get this result). However, in the case of the Triceratops fractal formula, this condition of |z|²>4 was not enough, so we excluded some points which might otherwise be parts of the fractal (and be black). Increasing that to e.g. 32, we might get a fractal very similar to the Mandelbrot set:

The f(z)=(z²+z)/2+c fractal with escape condition |z|²>32

Zooming in this fractal makes images similar to those of the Mandelbrot set... and conversely, if we reduce the escape condition of the Mandelbrot fractal (to e.g. 1 or 0.8), we may get images similar to those of the Triceratops fractal... Thus the secret of these interesting images lies in the escape condition, not really in the formula.

Zoom in the Mandelbrot fractal but with escape condition |z|²>0.7

Proof of an interesting statement about the sums of two squares

In an earlier blog post I have mentioned a conjecture of mine without proof:

"If a number is the sum of two consecutive square numbers, and it is not sum of any other two square numbers, then the number is prime. And in reverse, if the sum of two consecutive square numbers is prime, then it is not the sum of two other square numbers (where 0 is considered a square number)."

After I republished those earlier blog posts in this newer blog, that statement began to interest me again, so I took the time to think on it. I checked the conjecture with a C computer program for small numbers (i.e. until 50000²+50001²), and I also generalized the conjecture (and also checked that for some numbers) in the following way:

"Let S be the sum of an even and an odd square number relatively prime to each other. S is prime if and only if S is not sum of any other two square numbers (other than used in its definition)."

Here is some output of a C program I checked this conjecture with:

  2²+  1²=     5 (prime) (ok)
  4²+  1²=    17 (prime) (ok)
  4²+  3²=    25=   5*    5=   0²+   5²
  6²+  1²=    37 (prime) (ok)
  6²+  5²=    61 (prime) (ok)
  8²+  1²=    65=   5*   13=   4²+   7²
  8²+  3²=    73 (prime) (ok)
  8²+  5²=    89 (prime) (ok)
  8²+  7²=   113 (prime) (ok)
 10²+  1²=   101 (prime) (ok)
 10²+  3²=   109 (prime) (ok)
 10²+  7²=   149 (prime) (ok)
 10²+  9²=   181 (prime) (ok)
 12²+  1²=   145=   5*   29=   8²+   9²
 12²+  5²=   169=  13*   13=   0²+  13²
 12²+  7²=   193 (prime) (ok)
 12²+ 11²=   265=   5*   53=   3²+  16²
 14²+  1²=   197 (prime) (ok)
 14²+  3²=   205=   5*   41=   6²+  13²
 14²+  5²=   221=  13*   17=  10²+  11²
 14²+  9²=   277 (prime) (ok)
 14²+ 11²=   317 (prime) (ok)
 14²+ 13²=   365=   5*   73=   2²+  19²
 16²+  1²=   257 (prime) (ok)
 16²+  3²=   265=   5*   53=  11²+  12²
 16²+  5²=   281 (prime) (ok)
 16²+  7²=   305=   5*   61=   4²+  17²
 16²+  9²=   337 (prime) (ok)
 16²+ 11²=   377=  13*   29=   4²+  19²
 16²+ 13²=   425=   5*   85=   5²+  20²
 16²+ 15²=   481=  13*   37=   9²+  20²
 18²+  1²=   325=   5*   65=   6²+  17²
 18²+  5²=   349 (prime) (ok)
 18²+  7²=   373 (prime) (ok)
 18²+ 11²=   445=   5*   89=   2²+  21²
 18²+ 13²=   493=  17*   29=   3²+  22²
 18²+ 17²=   613 (prime) (ok)
 20²+  1²=   401 (prime) (ok)
 20²+  3²=   409 (prime) (ok)
 20²+  7²=   449 (prime) (ok)
 20²+  9²=   481=  13*   37=  15²+  16²
 20²+ 11²=   521 (prime) (ok)
 20²+ 13²=   569 (prime) (ok)
 20²+ 17²=   689=  13*   53=   8²+  25²
 20²+ 19²=   761 (prime) (ok)
 22²+  1²=   485=   5*   97=  14²+  17²
 22²+  3²=   493=  17*   29=  13²+  18²
 22²+  5²=   509 (prime) (ok)
 22²+  7²=   533=  13*   41=   2²+  23²
 22²+  9²=   565=   5*  113=   6²+  23²
 22²+ 13²=   653 (prime) (ok)
 22²+ 15²=   709 (prime) (ok)
 22²+ 17²=   773 (prime) (ok)
 22²+ 19²=   845=   5*  169=   2²+  29²
 22²+ 21²=   925=   5*  185=   5²+  30²
 24²+  1²=   577 (prime) (ok)
 24²+  5²=   601 (prime) (ok)
 24²+  7²=   625=   5*  125=   0²+  25²
 24²+ 11²=   697=  17*   41=  16²+  21²
 24²+ 13²=   745=   5*  149=   4²+  27²
 24²+ 17²=   865=   5*  173=   9²+  28²
 24²+ 19²=   937 (prime) (ok)
 24²+ 23²=  1105=   5*  221=   4²+  33²
 26²+  1²=   677 (prime) (ok)
 26²+  3²=   685=   5*  137=  18²+  19²
 26²+  5²=   701 (prime) (ok)
 26²+  7²=   725=   5*  145=  10²+  25²
 26²+  9²=   757 (prime) (ok)
 26²+ 11²=   797 (prime) (ok)
 26²+ 15²=   901=  17*   53=   1²+  30²
 26²+ 17²=   965=   5*  193=   2²+  31²
 26²+ 19²=  1037=  17*   61=  14²+  29²
 26²+ 21²=  1117 (prime) (ok)
 26²+ 23²=  1205=   5*  241=   7²+  34²
 26²+ 25²=  1301 (prime) (ok)
 28²+  1²=   785=   5*  157=  16²+  23²
 28²+  3²=   793=  13*   61=   8²+  27²
 28²+  5²=   809 (prime) (ok)
 28²+  9²=   865=   5*  173=  17²+  24²
 28²+ 11²=   905=   5*  181=   8²+  29²
 28²+ 13²=   953 (prime) (ok)
 28²+ 15²=  1009 (prime) (ok)
 28²+ 17²=  1073=  29*   37=   7²+  32²
 28²+ 19²=  1145=   5*  229=  11²+  32²
 28²+ 23²=  1313=  13*  101=  17²+  32²
 28²+ 25²=  1409 (prime) (ok)
 28²+ 27²=  1513=  17*   89=  12²+  37²
 30²+  1²=   901=  17*   53=  15²+  26²
 30²+  7²=   949=  13*   73=  18²+  25²
 30²+ 11²=  1021 (prime) (ok)
 30²+ 13²=  1069 (prime) (ok)
 30²+ 17²=  1189=  29*   41=  10²+  33²
 30²+ 19²=  1261=  13*   97=   6²+  35²
 30²+ 23²=  1429 (prime) (ok)
 30²+ 29²=  1741 (prime) (ok)
 32²+  1²=  1025=   5*  205=   8²+  31²
 32²+  3²=  1033 (prime) (ok)
 32²+  5²=  1049 (prime) (ok)
 32²+  7²=  1073=  29*   37=  17²+  28²
 32²+  9²=  1105=   5*  221=   4²+  33²
 32²+ 11²=  1145=   5*  229=  19²+  28²
 32²+ 13²=  1193 (prime) (ok)
 32²+ 15²=  1249 (prime) (ok)
 32²+ 17²=  1313=  13*  101=  23²+  28²
 32²+ 19²=  1385=   5*  277=   4²+  37²
 32²+ 21²=  1465=   5*  293=  13²+  36²
 32²+ 23²=  1553 (prime) (ok)
 32²+ 25²=  1649=  17*   97=   7²+  40²
 32²+ 27²=  1753 (prime) (ok)
 32²+ 29²=  1865=   5*  373=   4²+  43²
 32²+ 31²=  1985=   5*  397=   7²+  44²
 34²+  1²=  1157=  13*   89=  14²+  31²
 34²+  3²=  1165=   5*  233=  18²+  29²
 34²+  5²=  1181 (prime) (ok)
 34²+  7²=  1205=   5*  241=  23²+  26²
 34²+  9²=  1237 (prime) (ok)
 34²+ 11²=  1277 (prime) (ok)
 34²+ 13²=  1325=   5*  265=  10²+  35²
 34²+ 15²=  1381 (prime) (ok)
 34²+ 19²=  1517=  37*   41=  26²+  29²
 34²+ 21²=  1597 (prime) (ok)
 34²+ 23²=  1685=   5*  337=   2²+  41²
 34²+ 25²=  1781=  13*  137=  10²+  41²
 34²+ 27²=  1885=   5*  377=   6²+  43²
 34²+ 29²=  1997 (prime) (ok)
 34²+ 31²=  2117=  29*   73=   1²+  46²
 34²+ 33²=  2245=   5*  449=   6²+  47²
 36²+  1²=  1297 (prime) (ok)
 36²+  5²=  1321 (prime) (ok)
 36²+  7²=  1345=   5*  269=  16²+  33²
 36²+ 11²=  1417=  13*  109=  24²+  29²
 36²+ 13²=  1465=   5*  293=  21²+  32²
 36²+ 17²=  1585=   5*  317=   8²+  39²
 36²+ 19²=  1657 (prime) (ok)
 36²+ 23²=  1825=   5*  365=  12²+  41²
 36²+ 25²=  1921=  17*  113=  20²+  39²
 36²+ 29²=  2137 (prime) (ok)
 36²+ 31²=  2257=  37*   61=  24²+  41²
 36²+ 35²=  2521 (prime) (ok)
 38²+  1²=  1445=   5*  289=  17²+  34²
 38²+  3²=  1453 (prime) (ok)
 38²+  5²=  1469=  13*  113=  10²+  37²
 38²+  7²=  1493 (prime) (ok)
 38²+  9²=  1525=   5*  305=   2²+  39²
 38²+ 11²=  1565=   5*  313=  14²+  37²
 38²+ 13²=  1613 (prime) (ok)
 38²+ 15²=  1669 (prime) (ok)
 38²+ 17²=  1733 (prime) (ok)
 38²+ 21²=  1885=   5*  377=   6²+  43²
 38²+ 23²=  1973 (prime) (ok)
 38²+ 25²=  2069 (prime) (ok)
 38²+ 27²=  2173=  41*   53=  18²+  43²
 38²+ 29²=  2285=   5*  457=  13²+  46²
 38²+ 31²=  2405=   5*  481=   2²+  49²
 38²+ 33²=  2533=  17*  149=  18²+  47²
 38²+ 35²=  2669=  17*  157=  13²+  50²
 38²+ 37²=  2813=  29*   97=   2²+  53²
 40²+  1²=  1601 (prime) (ok)
 40²+  3²=  1609 (prime) (ok)
 40²+  7²=  1649=  17*   97=  25²+  32²
 40²+  9²=  1681=  41*   41=   0²+  41²
 40²+ 11²=  1721 (prime) (ok)
 40²+ 13²=  1769=  29*   61=  20²+  37²
 40²+ 17²=  1889 (prime) (ok)
 40²+ 19²=  1961=  37*   53=   5²+  44²
 40²+ 21²=  2041=  13*  157=   4²+  45²
 40²+ 23²=  2129 (prime) (ok)
 40²+ 27²=  2329=  17*  137=   5²+  48²
 40²+ 29²=  2441 (prime) (ok)
 40²+ 31²=  2561=  13*  197=  25²+  44²
 40²+ 33²=  2689 (prime) (ok)
 40²+ 37²=  2969 (prime) (ok)
 40²+ 39²=  3121 (prime) (ok)
 42²+  1²=  1765=   5*  353=  26²+  33²
 42²+  5²=  1789 (prime) (ok)
 42²+ 11²=  1885=   5*  377=   6²+  43²
 42²+ 13²=  1933 (prime) (ok)
 42²+ 17²=  2053 (prime) (ok)
 42²+ 19²=  2125=   5*  425=   3²+  46²
 42²+ 23²=  2293 (prime) (ok)
 42²+ 25²=  2389 (prime) (ok)
 42²+ 29²=  2605=   5*  521=   2²+  51²
 42²+ 31²=  2725=   5*  545=  15²+  50²
 42²+ 37²=  3133=  13*  241=  18²+  53²
 42²+ 41²=  3445=   5*  689=   9²+  58²
 44²+  1²=  1937=  13*  149=  16²+  41²
 44²+  3²=  1945=   5*  389=  24²+  37²
 44²+  5²=  1961=  37*   53=  19²+  40²
 44²+  7²=  1985=   5*  397=  31²+  32²
 44²+  9²=  2017 (prime) (ok)
 44²+ 13²=  2105=   5*  421=  16²+  43²
 44²+ 15²=  2161 (prime) (ok)
 44²+ 17²=  2225=   5*  445=   4²+  47²
 44²+ 19²=  2297 (prime) (ok)
 44²+ 21²=  2377 (prime) (ok)
 44²+ 23²=  2465=   5*  493=   8²+  49²
 44²+ 25²=  2561=  13*  197=  31²+  40²
 44²+ 27²=  2665=   5*  533=   8²+  51²
 44²+ 29²=  2777 (prime) (ok)
 44²+ 31²=  2897 (prime) (ok)
 44²+ 35²=  3161=  29*  109=   5²+  56²
 44²+ 37²=  3305=   5*  661=  13²+  56²
 44²+ 39²=  3457 (prime) (ok)
 44²+ 41²=  3617 (prime) (ok)
 44²+ 43²=  3785=   5*  757=   8²+  61²
 46²+  1²=  2117=  29*   73=  31²+  34²
 46²+  3²=  2125=   5*  425=  10²+  45²
 46²+  5²=  2141 (prime) (ok)
 46²+  7²=  2165=   5*  433=  22²+  41²
 46²+  9²=  2197=  13*  169=  26²+  39²
 46²+ 11²=  2237 (prime) (ok)
 46²+ 13²=  2285=   5*  457=  29²+  38²
 46²+ 15²=  2341 (prime) (ok)
 46²+ 17²=  2405=   5*  481=   2²+  49²
 46²+ 19²=  2477 (prime) (ok)
 46²+ 21²=  2557 (prime) (ok)
 46²+ 25²=  2741 (prime) (ok)
 46²+ 27²=  2845=   5*  569=   6²+  53²
 46²+ 29²=  2957 (prime) (ok)
 46²+ 31²=  3077=  17*  181=  26²+  49²
 46²+ 33²=  3205=   5*  641=  17²+  54²
 46²+ 35²=  3341=  13*  257=  29²+  50²
 46²+ 37²=  3485=   5*  697=   2²+  59²
 46²+ 39²=  3637 (prime) (ok)
 46²+ 41²=  3797 (prime) (ok)
 46²+ 43²=  3965=   5*  793=  11²+  62²
 46²+ 45²=  4141=  41*  101=  35²+  54²
 48²+  1²=  2305=   5*  461=  28²+  39²
 48²+  5²=  2329=  17*  137=  27²+  40²
 48²+  7²=  2353=  13*  181=  12²+  47²
 48²+ 11²=  2425=   5*  485=  20²+  45²
 48²+ 13²=  2473 (prime) (ok)
 48²+ 17²=  2593 (prime) (ok)
 48²+ 19²=  2665=   5*  533=   8²+  51²
 48²+ 23²=  2833 (prime) (ok)
 48²+ 25²=  2929=  29*  101=  15²+  52²
 48²+ 29²=  3145=   5*  629=   3²+  56²
 48²+ 31²=  3265=   5*  653=   4²+  57²
 48²+ 35²=  3529 (prime) (ok)
 48²+ 37²=  3673 (prime) (ok)
 48²+ 41²=  3985=   5*  797=   4²+  63²
 48²+ 43²=  4153 (prime) (ok)
 48²+ 47²=  4513 (prime) (ok)
 50²+  1²=  2501=  41*   61=  10²+  49²
 50²+  3²=  2509=  13*  193=  22²+  45²
 50²+  7²=  2549 (prime) (ok)
 50²+  9²=  2581=  29*   89=  30²+  41²
 50²+ 11²=  2621 (prime) (ok)
 50²+ 13²=  2669=  17*  157=  35²+  38²
 50²+ 17²=  2789 (prime) (ok)
 50²+ 19²=  2861 (prime) (ok)
 50²+ 21²=  2941=  17*  173=   5²+  54²
 50²+ 23²=  3029=  13*  233=   2²+  55²
 50²+ 27²=  3229 (prime) (ok)
 50²+ 29²=  3341=  13*  257=  35²+  46²
 50²+ 31²=  3461 (prime) (ok)
 50²+ 33²=  3589=  37*   97=  15²+  58²
 50²+ 37²=  3869=  53*   73=   5²+  62²
 50²+ 39²=  4021 (prime) (ok)
 50²+ 41²=  4181=  37*  113=  34²+  55²
 50²+ 43²=  4349 (prime) (ok)
 50²+ 47²=  4709=  17*  277=  22²+  65²
 50²+ 49²=  4901=  13*  377=   1²+  70²

(Edited.) NOTE: there can be other tests for the case when the odd number is greater than the even one:

  3²+  2²=    13 (prime) (ok)
  5²+  2²=    29 (prime) (ok)
  5²+  4²=    41 (prime) (ok)
  7²+  2²=    53 (prime) (ok)
  7²+  4²=    65=   5*   13=   1²+   8²
  7²+  6²=    85=   5*   17=   2²+   9²
  9²+  2²=    85=   5*   17=   6²+   7²
  9²+  4²=    97 (prime) (ok)
  9²+  8²=   145=   5*   29=   1²+  12²
 11²+  2²=   125=   5*   25=   5²+  10²
 11²+  4²=   137 (prime) (ok)
 11²+  6²=   157 (prime) (ok)
 11²+  8²=   185=   5*   37=   4²+  13²
 11²+ 10²=   221=  13*   17=   5²+  14²
 13²+  2²=   173 (prime) (ok)
 13²+  4²=   185=   5*   37=   8²+  11²
 13²+  6²=   205=   5*   41=   3²+  14²
 13²+  8²=   233 (prime) (ok)
 13²+ 10²=   269 (prime) (ok)
 13²+ 12²=   313 (prime) (ok)
 15²+  2²=   229 (prime) (ok)
 15²+  4²=   241 (prime) (ok)
 15²+  8²=   289=  17*   17=   0²+  17²
 15²+ 14²=   421 (prime) (ok)
 17²+  2²=   293 (prime) (ok)
 17²+  4²=   305=   5*   61=   7²+  16²
 17²+  6²=   325=   5*   65=   1²+  18²
 17²+  8²=   353 (prime) (ok)
 17²+ 10²=   389 (prime) (ok)
 17²+ 12²=   433 (prime) (ok)
 17²+ 14²=   485=   5*   97=   1²+  22²
 17²+ 16²=   545=   5*  109=   4²+  23²
 19²+  2²=   365=   5*   73=  13²+  14²
 19²+  4²=   377=  13*   29=  11²+  16²
 19²+  6²=   397 (prime) (ok)
 19²+  8²=   425=   5*   85=   5²+  20²
 19²+ 10²=   461 (prime) (ok)
 19²+ 12²=   505=   5*  101=   8²+  21²
 19²+ 14²=   557 (prime) (ok)
 19²+ 16²=   617 (prime) (ok)
 19²+ 18²=   685=   5*  137=   3²+  26²
 21²+  2²=   445=   5*   89=  11²+  18²
 21²+  4²=   457 (prime) (ok)
 21²+  8²=   505=   5*  101=  12²+  19²
 21²+ 10²=   541 (prime) (ok)
 21²+ 16²=   697=  17*   41=  11²+  24²
 21²+ 20²=   841=  29*   29=   0²+  29²
 23²+  2²=   533=  13*   41=   7²+  22²
 23²+  4²=   545=   5*  109=  16²+  17²
 23²+  6²=   565=   5*  113=   9²+  22²
 23²+  8²=   593 (prime) (ok)
 23²+ 10²=   629=  17*   37=   2²+  25²
 23²+ 12²=   673 (prime) (ok)
 23²+ 14²=   725=   5*  145=   7²+  26²
 23²+ 16²=   785=   5*  157=   1²+  28²
 23²+ 18²=   853 (prime) (ok)
 23²+ 20²=   929 (prime) (ok)
 23²+ 22²=  1013 (prime) (ok)
 25²+  2²=   629=  17*   37=  10²+  23²
 25²+  4²=   641 (prime) (ok)
 25²+  6²=   661 (prime) (ok)
 25²+  8²=   689=  13*   53=  17²+  20²
 25²+ 12²=   769 (prime) (ok)
 25²+ 14²=   821 (prime) (ok)
 25²+ 16²=   881 (prime) (ok)
 25²+ 18²=   949=  13*   73=   7²+  30²
 25²+ 22²=  1109 (prime) (ok)
 25²+ 24²=  1201 (prime) (ok)
 27²+  2²=   733 (prime) (ok)
 27²+  4²=   745=   5*  149=  13²+  24²
 27²+  8²=   793=  13*   61=   3²+  28²
 27²+ 10²=   829 (prime) (ok)
 27²+ 14²=   925=   5*  185=   5²+  30²
 27²+ 16²=   985=   5*  197=  12²+  29²
 27²+ 20²=  1129 (prime) (ok)
 27²+ 22²=  1213 (prime) (ok)
 27²+ 26²=  1405=   5*  281=   6²+  37²
 29²+  2²=   845=   5*  169=  13²+  26²
 29²+  4²=   857 (prime) (ok)
 29²+  6²=   877 (prime) (ok)
 29²+  8²=   905=   5*  181=  11²+  28²
 29²+ 10²=   941 (prime) (ok)
 29²+ 12²=   985=   5*  197=  16²+  27²
 29²+ 14²=  1037=  17*   61=  19²+  26²
 29²+ 16²=  1097 (prime) (ok)
 29²+ 18²=  1165=   5*  233=   3²+  34²
 29²+ 20²=  1241=  17*   73=   4²+  35²
 29²+ 22²=  1325=   5*  265=  10²+  35²
 29²+ 24²=  1417=  13*  109=  11²+  36²
 29²+ 26²=  1517=  37*   41=  19²+  34²
 29²+ 28²=  1625=   5*  325=   5²+  40²
 31²+  2²=   965=   5*  193=  17²+  26²
 31²+  4²=   977 (prime) (ok)
 31²+  6²=   997 (prime) (ok)
 31²+  8²=  1025=   5*  205=   1²+  32²
 31²+ 10²=  1061 (prime) (ok)
 31²+ 12²=  1105=   5*  221=   4²+  33²
 31²+ 14²=  1157=  13*   89=   1²+  34²
 31²+ 16²=  1217 (prime) (ok)
 31²+ 18²=  1285=   5*  257=  14²+  33²
 31²+ 20²=  1361 (prime) (ok)
 31²+ 22²=  1445=   5*  289=   1²+  38²
 31²+ 24²=  1537=  29*   53=   4²+  39²
 31²+ 26²=  1637 (prime) (ok)
 31²+ 28²=  1745=   5*  349=   8²+  41²
 31²+ 30²=  1861 (prime) (ok)
 33²+  2²=  1093 (prime) (ok)
 33²+  4²=  1105=   5*  221=   9²+  32²
 33²+  8²=  1153 (prime) (ok)
 33²+ 10²=  1189=  29*   41=  17²+  30²
 33²+ 14²=  1285=   5*  257=  18²+  31²
 33²+ 16²=  1345=   5*  269=   7²+  36²
 33²+ 20²=  1489 (prime) (ok)
 33²+ 26²=  1765=   5*  353=   1²+  42²
 33²+ 28²=  1873 (prime) (ok)
 33²+ 32²=  2113 (prime) (ok)
 35²+  2²=  1229 (prime) (ok)
 35²+  4²=  1241=  17*   73=  20²+  29²
 35²+  6²=  1261=  13*   97=  19²+  30²
 35²+  8²=  1289 (prime) (ok)
 35²+ 12²=  1369=  37*   37=   0²+  37²
 35²+ 16²=  1481 (prime) (ok)
 35²+ 18²=  1549 (prime) (ok)
 35²+ 22²=  1709 (prime) (ok)
 35²+ 24²=  1801 (prime) (ok)
 35²+ 26²=  1901 (prime) (ok)
 35²+ 32²=  2249=  13*  173=  20²+  43²
 35²+ 34²=  2381 (prime) (ok)
 37²+  2²=  1373 (prime) (ok)
 37²+  4²=  1385=   5*  277=  19²+  32²
 37²+  6²=  1405=   5*  281=  26²+  27²
 37²+  8²=  1433 (prime) (ok)
 37²+ 10²=  1469=  13*  113=   5²+  38²
 37²+ 12²=  1513=  17*   89=  27²+  28²
 37²+ 14²=  1565=   5*  313=  11²+  38²
 37²+ 16²=  1625=   5*  325=   5²+  40²
 37²+ 18²=  1693 (prime) (ok)
 37²+ 20²=  1769=  29*   61=  13²+  40²
 37²+ 22²=  1853=  17*  109=   2²+  43²
 37²+ 24²=  1945=   5*  389=   3²+  44²
 37²+ 26²=  2045=   5*  409=  14²+  43²
 37²+ 28²=  2153 (prime) (ok)
 37²+ 30²=  2269 (prime) (ok)
 37²+ 32²=  2393 (prime) (ok)
 37²+ 34²=  2525=   5*  505=   5²+  50²
 37²+ 36²=  2665=   5*  533=   8²+  51²
 39²+  2²=  1525=   5*  305=   9²+  38²
 39²+  4²=  1537=  29*   53=  24²+  31²
 39²+  8²=  1585=   5*  317=  17²+  36²
 39²+ 10²=  1621 (prime) (ok)
 39²+ 14²=  1717=  17*  101=   6²+  41²
 39²+ 16²=  1777 (prime) (ok)
 39²+ 20²=  1921=  17*  113=  25²+  36²
 39²+ 22²=  2005=   5*  401=  18²+  41²
 39²+ 28²=  2305=   5*  461=   1²+  48²
 39²+ 32²=  2545=   5*  509=  12²+  49²
 39²+ 34²=  2677 (prime) (ok)
 39²+ 38²=  2965=   5*  593=   7²+  54²
 41²+  2²=  1685=   5*  337=  23²+  34²
 41²+  4²=  1697 (prime) (ok)
 41²+  6²=  1717=  17*  101=  14²+  39²
 41²+  8²=  1745=   5*  349=  28²+  31²
 41²+ 10²=  1781=  13*  137=  25²+  34²
 41²+ 12²=  1825=   5*  365=  15²+  40²
 41²+ 14²=  1877 (prime) (ok)
 41²+ 16²=  1937=  13*  149=   1²+  44²
 41²+ 18²=  2005=   5*  401=  22²+  39²
 41²+ 20²=  2081 (prime) (ok)
 41²+ 22²=  2165=   5*  433=   7²+  46²
 41²+ 24²=  2257=  37*   61=  31²+  36²
 41²+ 26²=  2357 (prime) (ok)
 41²+ 28²=  2465=   5*  493=   8²+  49²
 41²+ 30²=  2581=  29*   89=   9²+  50²
 41²+ 32²=  2705=   5*  541=   1²+  52²
 41²+ 34²=  2837 (prime) (ok)
 41²+ 36²=  2977=  13*  229=  24²+  49²
 41²+ 38²=  3125=   5*  625=  10²+  55²
 41²+ 40²=  3281=  17*  193=  16²+  55²
 43²+  2²=  1853=  17*  109=  22²+  37²
 43²+  4²=  1865=   5*  373=  29²+  32²
 43²+  6²=  1885=   5*  377=  11²+  42²
 43²+  8²=  1913 (prime) (ok)
 43²+ 10²=  1949 (prime) (ok)
 43²+ 12²=  1993 (prime) (ok)
 43²+ 14²=  2045=   5*  409=  26²+  37²
 43²+ 16²=  2105=   5*  421=  13²+  44²
 43²+ 18²=  2173=  41*   53=  27²+  38²
 43²+ 20²=  2249=  13*  173=  32²+  35²
 43²+ 22²=  2333 (prime) (ok)
 43²+ 24²=  2425=   5*  485=  11²+  48²
 43²+ 26²=  2525=   5*  505=   5²+  50²
 43²+ 28²=  2633 (prime) (ok)
 43²+ 30²=  2749 (prime) (ok)
 43²+ 32²=  2873=  13*  221=   8²+  53²
 43²+ 34²=  3005=   5*  601=  14²+  53²
 43²+ 36²=  3145=   5*  629=   3²+  56²
 43²+ 38²=  3293=  37*   89=  22²+  53²
 43²+ 40²=  3449 (prime) (ok)
 43²+ 42²=  3613 (prime) (ok)
 45²+  2²=  2029 (prime) (ok)
 45²+  4²=  2041=  13*  157=  21²+  40²
 45²+  8²=  2089 (prime) (ok)
 45²+ 14²=  2221 (prime) (ok)
 45²+ 16²=  2281 (prime) (ok)
 45²+ 22²=  2509=  13*  193=   3²+  50²
 45²+ 26²=  2701=  37*   73=  10²+  51²
 45²+ 28²=  2809=  53*   53=   0²+  53²
 45²+ 32²=  3049 (prime) (ok)
 45²+ 34²=  3181 (prime) (ok)
 45²+ 38²=  3469 (prime) (ok)
 45²+ 44²=  3961=  17*  233=  19²+  60²
 47²+  2²=  2213 (prime) (ok)
 47²+  4²=  2225=   5*  445=  17²+  44²
 47²+  6²=  2245=   5*  449=  33²+  34²
 47²+  8²=  2273 (prime) (ok)
 47²+ 10²=  2309 (prime) (ok)
 47²+ 12²=  2353=  13*  181=   7²+  48²
 47²+ 14²=  2405=   5*  481=   2²+  49²
 47²+ 16²=  2465=   5*  493=   8²+  49²
 47²+ 18²=  2533=  17*  149=  33²+  38²
 47²+ 20²=  2609 (prime) (ok)
 47²+ 22²=  2693 (prime) (ok)
 47²+ 24²=  2785=   5*  557=   9²+  52²
 47²+ 26²=  2885=   5*  577=  22²+  49²
 47²+ 28²=  2993=  41*   73=  17²+  52²
 47²+ 30²=  3109 (prime) (ok)
 47²+ 32²=  3233=  53*   61=  23²+  52²
 47²+ 34²=  3365=   5*  673=   1²+  58²
 47²+ 36²=  3505=   5*  701=  16²+  57²
 47²+ 38²=  3653=  13*  281=  17²+  58²
 47²+ 40²=  3809=  13*  293=  28²+  55²
 47²+ 42²=  3973=  29*  137=   2²+  63²
 47²+ 44²=  4145=   5*  829=   7²+  64²
 47²+ 46²=  4325=   5*  865=  10²+  65²
 49²+  2²=  2405=   5*  481=  14²+  47²
 49²+  4²=  2417 (prime) (ok)
 49²+  6²=  2437 (prime) (ok)
 49²+  8²=  2465=   5*  493=  16²+  47²
 49²+ 10²=  2501=  41*   61=   1²+  50²
 49²+ 12²=  2545=   5*  509=  32²+  39²
 49²+ 16²=  2657 (prime) (ok)
 49²+ 18²=  2725=   5*  545=  15²+  50²
 49²+ 20²=  2801 (prime) (ok)
 49²+ 22²=  2885=   5*  577=  26²+  47²
 49²+ 24²=  2977=  13*  229=  36²+  41²
 49²+ 26²=  3077=  17*  181=  31²+  46²
 49²+ 30²=  3301 (prime) (ok)
 49²+ 32²=  3425=   5*  685=  17²+  56²
 49²+ 34²=  3557 (prime) (ok)
 49²+ 36²=  3697 (prime) (ok)
 49²+ 38²=  3845=   5*  769=   1²+  62²
 49²+ 40²=  4001 (prime) (ok)
 49²+ 44²=  4337 (prime) (ok)
 49²+ 46²=  4517 (prime) (ok)
 49²+ 48²=  4705=   5*  941=   9²+  68²
 51²+  2²=  2605=   5*  521=  29²+  42²
 51²+  4²=  2617 (prime) (ok)
 51²+  8²=  2665=   5*  533=  19²+  48²
 51²+ 10²=  2701=  37*   73=  26²+  45²
 51²+ 14²=  2797 (prime) (ok)
 51²+ 16²=  2857 (prime) (ok)
 51²+ 20²=  3001 (prime) (ok)
 51²+ 22²=  3085=   5*  617=  13²+  54²
 51²+ 26²=  3277=  29*  113=  19²+  54²
 51²+ 28²=  3385=   5*  677=  24²+  53²
 51²+ 32²=  3625=   5*  725=   5²+  60²
 51²+ 38²=  4045=   5*  809=  18²+  61²
 51²+ 40²=  4201 (prime) (ok)
 51²+ 44²=  4537=  13*  349=  21²+  64²
 51²+ 46²=  4717=  53*   89=  19²+  66²
 51²+ 50²=  5101 (prime) (ok)

Thus the time has come to prove this conjecture mathematically... At first, we suppose that S is the sum of two "other" square numbers, and we prove that it comes from this that S is not prime. So our supposition is:

S=(2k)²+(2j+1)²=(2m)²+(2n+1)² 

gcd(2k,2j+1)=1

First we convert the first equation to the following form:

(x-y)²+(z+v)²=(x+y)²+(z-v)²

where (z+v) is the greatest of the terms (2k), (2j+1), (2m) and (2n+1), and (z-v) is the other term which has the same parity as (z+v). This can be done, as (say) the following system of equations has one solution:

x-y=2k

z+v=2j+1

x+y=2m

z-v=2n+1

And it can be solved easily for x,y,z,v (they can be given in such a way that x>=1, z>=1,y>=1,v>=1, if -say- m>k and j>n):

x=k+m

z=j+n+1

y=m-k

v=j-n

Thus we can work with the variables x,y,z,v from now on. We can transform our equation in the following way:

(x-y)²+(z+v)²=(x+y)²+(z-v)²

x²+y²+z²+v²-2xy+2zv=x²+y²+z²+v²+2xy-2zv

4zv=4xy

zv=xy

... and using this identity, we can substitute it back into our equation, we get:

S=(x-y)²+(z+v)²=(x+y)²+(z-v)²=x²+y²+z²+v²

But it's still more interesting to us that T=z*v=x*y. In the prime factors of this product (T), some are element(s) of the set of factors of z, some are element(s) of the set of factors of v... and some are of x and some are of y. These four sets intersect in four subsets, which may be called a, b, c and d:

Thus we can write (say) the following system of equations:

x=ac

y=bd

z=ad

v=bc

Afterwards, we can square all equations the following way:

x²=a²c²

y²=b²d²

z²=a²d²

v²=b²c²

Adding them together gives:

S=x²+y²+z²+v²=a²c²+b²d²+a²d²+b²c²=(a²+b²)(c²+d²)=S

And we can reckon that this is equal to the sum S. As a, b, c and d are at least 1, S is a composite number, not prime. I've used the Brahmagupta-Fibonacci identity (otherwise called the Diophantus identity) when proving this part of the conjecture. (We can use the rule of contraposition to show that if S is prime, then it is not the sum of two other square numbers.) Note: the Brahmagupta-Fibonacci identity is this (can be proven by simple algebra):

(a²+b²)(c²+d²)=(ac-bd)²+(ad+bc)²=(ac+bd)²+(ad-bc)²

The other part of the conjecture seemed to be more tricky. This says that if S is not prime, then S is the sum of two other square numbers. Thus the following is supposed:

S=(2k)²+(2j+1)²=e*f

gcd(2k,2j+1)=1

 e>1,f>1,k>=1,j>=0

... and we must prove that e*f is of the form:

e=a²+b² 

f=c²+d²

If we managed to prove this, it would come from the "Brahmagupta-Fibonacci identity" that S can be expressed as the sums of two squares two ways, one of which may be different from (2k)²+(2j+1)², proving this part of the conjecture. Now what? :-) I've found some interesting theorems in WikiPedia that are related. These are the following:

• Fermat's theorem on sums of two squares (with proof)
• Sum of two squares theorem
• and of course, the Brahmagupta-Fibonacci identity

Now what? :-) The "Sum of two squares theorem" says that:

An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no prime congruent to 3 modulo 4 raised to an odd power.

(https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem#cite_note-1
Underwood Dudley (1978). Elementary Number Theory (2 ed.). W.H. Freeman and Company.)

Now S can be written as a sum of two squares, so in its prime decomposition all possible primes of the form (4p+3) are squared. Let the product of all of these be s². If s² is not a divisor of either e or f, then let's redefine e and f in such a way that e should be divisible by s² and f should also have some divisors of the form (4p+1). (If all that we have were only s², then S would be the sum of s² and 0², proving our conjecture, or if this were the same as our initial condition, then k>=1 would not hold.) According to "Fermat's theorem on sums of two squares", all the other primes of the form (4p+1) can be expressed as t²+u². According to the Brahmagupta-Fibonacci identity, when we multiply one such prime with another, the result can also be expressed as t²+u². In summary, this makes e of the form e=s²t²+s²u² and f=q²+r². Thus we have reached our desired goal that:

e=a²+b² 

f=c²+d²

... and the Brahmagupta-Fibonacci identity proves the second part of the conjecture. (We can use the rule of contraposition to show that if S is not the sum of two other square numbers, then the number is prime.)

There is another interesting theorem worth mentioning here, Lagrange's four-square theorem. I hoped that it might help proving the Goldbach conjecture (with the result in this blog post), but it turned out not to be that case.

(Edited.) NOTE: The conjecture is not true when one of the square numbers is 0. I've discovered it when checking my post, and after modifying my C program I also get some counterexamples:

  3²+  0²=     9=   3*    3 (ok)
  7²+  0²=    49=   7*    7 (ok)
  9²+  0²=    81=   3*   27 (ok)
 11²+  0²=   121=  11*   11 (ok)