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), and I printed them. 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)

I still had things to show from my math past

Part of this blog post is deleted... there were a lot of pictures here that are lost when republishing... however, here are two interesting statements worth sharing...

Statement 1:
We can make convex polyhedra from paper if we first construct their faces on paper, which meet at its vertices, and if we count those angles at these vertices that cover the areas which are NOT part of the surrounding faces (in theory), then we get 720 degrees.

Sketch of proof to Statement 1:
If F is the number of faces of the polyhedron, V is the number of vertices, and E is the number of edges, and N(x) is the number of vertices of face x, then we can say the following:

• The sum of inner angles of face x is (N(x)-2)·180 degrees, this is even taught in public education

• The sum of all inner angles of all faces is 180·((N(1)-2)+(N(2)-2)+(N(3)-2)+...+(N(F)-2))

• This is equal to 180·(2·E-2·F), or 360·(E-F)

• Checking out Euler's polyhedron theorem for convex polyhedra

• Euler's theorem can be used here, so this is equal to 360·(V-2)

• All the degrees at the vertices: 360·V

• So, all the degrees at the vertices minus the angles of the polyhedra at the vertices is 360·V-360·(V-2)=360·2=720 degrees, this is what our statement is about.

Edit: Since then, I learnt that this thing is already known, and called Polyhedral Curvature, and proven in a different way, which is also sketched on the AMS website.

Statement 2:
The statement of the (strong) Goldbach conjecture ("every even number greater than 2 can be written as the sum of two primes") is equivalent to the following statement: "For N > 1 natural number, every even number greater than N^2+1 can be written as the sum of two numbers that are greater than 1 and relatively prime to N!". See this link for more.

Sketch of proof to Statement 2:
Let P(i) be the "i"th prime number. If "Every even number greater than 1+P(i)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1), P(2), P(3), ..., P(i)", then let's write this condition for each of the primes: 

• Every even number greater than 1+P(1)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1)
• Every even number greater than 1+P(2)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1), P(2)
• Every even number greater than 1+P(3)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1), P(2), P(3)
• ...

• Every even number greater than 1+P(i)^2 can be written as the sum of two numbers that are greater than 1 and not divisible by P(1), P(2), P(3), ..., P(i) 

Of course, if the Goldbach conjecture is true, then all of these little statements are true, so our general statement is true as well, because primes are also relative primes at the same time. But does the converse hold? Or in other words, can these sums of numbers be sums of primes all the time, by choosing the right substatement?
For a number to be prime, it should not be divisible by any number less than its square root, except 1. So, if these two numbers are not divisible by any of P(1), P(2), P(3), ..., P(i), then consequently, their value is at least P(i+1), and if they are not prime (necessary for a counterexample), then their value is at least P(i+1)^2, altogether at least P(i+1)^2+P(i+1). If all the substatements in the above example are true until P(i), then the Goldbach conjecture is true from 1+P(i)^2+1 (and therefore, from 1+P(1)^2+1) until P(i+1)^2+P(i+1)-1 (at least, and below). Since we suppose all of them to be true towards 1+P(Infinity)^2, the Goldbach conjecture also comes from this sequence of substatements (i.e. true for even numbers 4-11, 11-29, 27-55, 51-etc.), which is otherwise expressed as a general statement we have proved now to be equivalent to the Goldbach conjecture.

The truth of the statement for non-primes is trivial, if it is true for P(i), it is also true for P(i+1) > N > P(i), because the statement for P(i) implies this.

I had a Hungarian-language HTML-language blog in 2009-2010

And I'm going to share the essence of it here...

• My Golden Sierpinski Triangle (Félegyházi háromszög) was already there.
• My Sharpening of the Goldbach conjecture was already there.
• Some of my thoughts about material implication were already there (imperfectly), including the following logical paradoxon: "If I were half as clever as a 14-year-old as I had been then, I would have been more clever than Albert Einstein when discovering the Theory of Relativity." (as it is said that everything comes from a false statement, and I was not half as clever).
• For each vertex of a convex polyhedron, take the angle that is left out when summing its angles at that vertex (so  substract the sum from 360 degrees), and prove that the sum of the  angles of this kind are equal to 720 degrees. (Note: I made polyhedra  from paper using this knowledge, so I didn't need advanced math.)
• For the segments between four points in 3D space, find a formula or  more formulas that sufficiently describe the constraints about the lengths of  these segments can be (and those will be more strict than the triangle inequalities, and maybe more strict than other known inequalities as well.
• Promotion of  Hungarian sources were already there (KÖMAL, books of George Pólya)
• Promotion of interesting things were there [cos(pi/5), n|(n-2)!-1, n!m!|(n+m-1)!]
• Some minor things were already there (Sokoban levels, my goals, etc)
• Maybe the following insight was not part of that blog, but an earlier homepage in the University: "Prove Fermat's Little Theorem by searching for the answer to the question: how much information can you store in N bits or units, if the bits are arranged around a circle and the places at the circle are indistinguishable?" (Note: I claim that I have rediscovered Fermat's Little Theorem before we learnt about it in the University, but you don't need to believe me, of course.)
• The following conjecture, which might be very significant: "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)." I even conjectured that this may be proved by the Brahmagupta-Fibonacci identity, but I did not have time for it.
• Transformation of the Collatz-conjecture to four 8k+2n+1 cases, where 0<=n<=3
• Trying to find a fractal similar to Barnsley, but symmetric (sorry, the images are lost before 2020).

That homepage of mine had many versions, I've found this much today. But if I'm sharing my old things anyway, why not share those things which were published in 2014, but removed?

• It was not exactly written by these words, but playing the Chaos-game with a ratio of 1/2 makes the Sierpinski-triangle, with the Golden Ratio, my Golden Sierpinski fractal imaged below, and there exist other ratios as well between the two, for which the overlapping of the fractal is lucky, because the overlapping parts are identical.

• Binary representation of rational numbers without decimal point, e.g.:

+1=01

+2=011

+4=0111

+3 = 01101

+3.141601562 = 011010010010001

• Optimize (maximize): O=L^3*W*P^2, where L+W+P=1
• Two very interesting (but secret) things in Planar Geometry

So this blog post was about my old things in bulk. Now comes a longer silence here, probably.

Searching for alternatives to material implication

This journey of my research has begun when we learnt about material implication in the University of Szeged. There was an exercise in the Discrete Mathematics book that we were solving on a lesson, and I did not agree with the official solution. I am probably not allowed to copy the example here, but I can share a sentence with a similar structure:

- Suppose that Andrew says that his hat is red only if Adam's hat is green. But Adam says that if his own hat is not green, then Andrew's hat is red. Can both of them be right?

If we use the truth table of material implication for conditional statements, and we use the same thing for "only if" constructs, then it's possible that both of them say the truth. My natural brain, however, felt that it is too strange. That's why I spent some time thinking on this occasionally, with mixed results. During my research, I have come across another strange example:

- Suppose that Andrew knows that there is no electricity supply, and he also knows that if there is no electricity, the lights cannot shine. Adam, however, knows that the switch is off. Is the following sentence true or false: "If the switch is on, the lights shine"?

According to the truth table of material implication Adam immediately knows that the sentence is true, for "everything comes from a false statement". This means that the sentence should be true according to Andrew's knowledge as well. According to the truth table of material implication this is OK, because we can suppose that the switch is never on. Yet here we can also feel something strange.

Many Logicians and Mathematicans have also had strange feelings about material implication. That's why some of them defined other kinds of implication, like indicative conditional, strict conditional, entailment, or even theories like relevant logic, and some others have written publications about the issue, like William Kneale and Martha Kneale wrote The Development of Logic. There's an entire chapter in it about the implication relation, about the ancient disputes of Philo the Dialectician, supporter of material implication against Diodorus Cronus, supporter of a stricter definition. The Development of Logic concluded that the truth table of a better implication should not fix the truth value of the implication in case the first sub-statement is false, like on the pictures. I share two pictures instead of one because I don't remember exactly what was in the book, but these two pictures are equivalent.

There was a time when I did not stop here. I had the following suppositions:

- Every true statements come from a true statement.

- Every true statements come from a false statement.

- A true statement comes from everything.

- A false statement does not necessarily come from another false statement.

Based on these premises, I could improve the truth table in a following way:

For some time, I believed that I've discovered the real implication here. There was another reason of thinking this: this way I could exclude the equivalence of A=1 and A=2 if A=3 and other things are unknown. Then I promoted this on a forum about philosophy, and people have attacked it, of course. They did not manage to disprove my arguments, but they have stimulated my thoughts in a way that enabled me to believe that I can disprove my arguments myself. In the antiquity, people did not use truth tables, but they used syllogism!

It was challenging to try to exclude the case of (false, false, false) in the truth table of implication, and in fact, this is possible by syllogisms...

1. Modus Tollens is true always:

If (If B Then A) and (Not A) Then (Not B)

2. The following is the only one assumption:
If B Then A

3. If the assumption and Modus Tollens are true, then this is true:
If True and (Not A) Then (Not B)

4. So if the assumption is true, then this is true:
If (Not A) Then (Not B)

5. Thus the following is always true:
If (If B Then A) Then (If (Not A) Then (Not B)).

6. Thus the rule of contraposition is correct.

7. If C is false and D is false, (Not C) is true and (Not D) is true, and "If (Not C) then (Not D)" is true.

8. From the previous assumptions in (7), the rule of contraposition says that "If (Not (Not D)) then (Not (Not C))" is true, in other words, "If D then C" is true.

9. This also means that if A is false and B is false, then "If A Then B" is true.

Thus we can use the rule of contraposition to exclude the other extra row from the truth table as well, so we get the truth table of material implication we learn about in schools!

At least they could provide a proof like this to it when we learn about it... but wait... the strange feelings about the material implication are still not resolved! How to resolve them? This is your homework! (Just kidding.) More seriously, I really feel that these strange feelings can be justified one day, when the knowledge of humankind advances... maybe I'll publish something else about in this blog (or elsewhere) later.

Currently, I conjecture that the solution would be about undetermined truth values of statements, which is against the materialistic worldview, but still a possibility. This means, for example, that something in the future can be either true or false, so not predestined. At least, it seems that these strange feelings of our soul come from an unconscious faith in this freedom, which has implications on our view of logical statements.

Sharpening the Goldbach conjecture

I like to think on famous mathematical conjectures, because these are real problems and their solution may be rewarded by fame. Even if I do not come to any solutions, I get some experience that is probably more valuable than the solution of made-up exercises. For example, I have already spent time trying to prove the Hadwiger conjecture in graph theory (including the Four-color conjecture, which is called theorem since), the Collatz conjecture, the Twin Prime conjecture, Goldbach's conjecture, Beal's conjecture, the Riemann hypothesis and the P versus NP problem, and as well as some own problems I came across during recreation.

From all of these conjectures, the Goldbach conjecture was the most interesting to me so far. This says that for every even number greater than 2, there exist two prime numbers such that the sum of these prime numbers are equal to the even number. You can probably see a picture about examples here for even numbers 4 to 500:

If we choose the prime numbers like I did on this picture, so that the prime numbers should have the least difference, we can notice that these prime numbers are all between the 1/4 and 3/4 of the original even number, and as the even numbers grow, the smaller prime numbers are not needed any more. This is actually my new conjecture that I called the Sharp Goldbach Conjecture, and I have checked it by computer for "small" even numbers (as far as I remember, some millions). By the way, this name for the new conjecture is probably not good, as I guess mathematicans call something sharp if it cannot be sharpened further. I cannot call it Strong Goldbach Conjecture either, because the Strong Goldbach Conjecture is the Goldbach conjecture itself for historical reasons, and Goldbach's Weak Conjecture is about any odd number greater than 5 being the sum of three prime numbers. I think I will call it my Sharpened Goldbach Conjecture.

Otherwise, as far as I remember, I showed to myself that the Goldbach conjecture itself is also equivalent to a conjecture that deals with only prime numbers greater than an arbitrary number, provided that the sum is also greater than a specific number (dependent on the arbitrary number). As far as I remember, I transformed the statements telescopically, and there was a square of a prime, but I do not remember exactly how. I tried to reproduce the proof here, while writing this blog post, but it turned out that I could not. So... this is your homework! (Just kidding.)

After these insights, I tried to prove the conjecture by using numbers NOT divisible by little numbers / primes instead of using prime numbers. This attempt did not succeed, but I realized that this is looking like Sieve Theory, by which other mathematicans may try to prove the Goldbach conjecture or the Twin Prime conjecture. So I think there might have been some chance for the proof, but I had not only mathematical problems to think on, but also real-life problems in the world.

Summary: My Sharpened Goldbach Conjecture says that every even integer greater than 3 can be written as the sum of two prime numbers between the 1/4 and 3/4 of the even integer. Sharpening conjectures might help in knowing their nature.