23/10/2022

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 Mandelbrot set

The Triceratops fractal
The Triceratops fractal

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

Triceratops fractal - picture 1

Triceratops fractal - picture 2

Triceratops fractal - picture 3

Triceratops fractal - picture 4

Triceratops fractal - picture 5

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 exit condition |z|²>32
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
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:
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)
  • 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.
Is this the truth table of the best implication?
Is this the truth table of the best implication?

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:

Can this be the truth table of a better implication?
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!
The truth table of material implication

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.

21/10/2022

I didn't win on the Alpine Fellowship Writing Prize, 2021 ("Untamed: On Wilderness and Civilization")

I didn't win on the Alpine Fellowship Writing Prize, 2021 (theme: "Untamed: On Wilderness and Civilization"). Here is my essay:

 

Civilization is the new Wilderness


As far as we know, our species (Homo Sapiens) has come into being in small communities. They were hunter-gatherers (like some monkeys), and there was a limited supply of food for them, that's why their population could not grow significantly. Some people call their system „primitive communism”, as they shared the food among themselves, and their land was common. This means that there was probably not much competition inside their own group, but instead, they were fighting or competing against other small groups. The soul of humankind has formed in this kind of environment, with its ability to love and hate, to forgive or avenge, to be brave or cowardly, to think, speak, act or omit. Later in history, agriculture allowed the sizes of populations to increase, so the smaller groups grew into city states and even nation states. This way they had more chance against smaller groups in wars, and this kind of civilization conquered the world (or at least, the Earth, if extraterrestials exist). As the magnitude of communities was increasing, love and hate, gratitude and retribution were no longer able to govern human relationships, so laws were required, like the Code of Hammurabi in Babylonia or the Ten Commandments of the Bible. These laws were usually able to sustain peace inside the states (preventing civil wars), but did not manage to sustain „primitive communism”. Inheritable social inequalities appeared, and people possessing military power (like the aristocrats in ancient Greece) or spiritual power, real or alleged (like the Brahmins in India) owned and commanded more. The institution of slavery in antiquity meant an even larger kind of inequality. Due to the unjust inequalities in society, more rivalries came up among members of the same state (e.g. „class struggles”, as some people call them, or competitions for jobs), so in case of humankind, the evolutionary struggle for existence (as Darwinism teaches) got two dimensions (not to mention the environment): struggle among states (or nations) and struggle among individuals of the same state (or nation). Well, this might have been so in prehistoric times as well, but on a different level. In today's world (the era of United Nations) there is less competition among groups (or nations), and more competition among individuals (or nuclear families). This means that we are more like bears and less like ants than before. Thus, it seems that we have been untamed!

Well, of course, the case is not as simple as that. People are still urged to be law-abiding, useful to others, and popular (to have many friends or acquaintances). There is just more individualism and cosmopolitanism on Earth than before. The point is that in today's world, civilization to humans is like wilderness to bears. It is no longer a strong alliance, but only an environment of rules telling how to compete. These rules or laws cannot ensure that people remain tame inside, as there are always loopholes or lacunae in such a system. For example, the conduct of pupils and students is worsening in schools, and the movies are more and more violent. There is cyberbullying despite the countermeasures. There are many video games with protagonists who actually commit crime in their imaginary world. There are soccer fans (or football hooligans) who might break shop-windows after a match. Copyright infringement is widespread, and there are also many other kinds of crime and corruption. Thus, it seems that many people in our era are born to be wild, which they remain to be inside, and only look tame from the outside as they get more sophisticated. It also seems that Christianity knew this, and tried to improve the situation by focusing on love (or alliance) instead of the laws. Still, the evolutionary struggle continues, and it still means competition among individuals (or nuclear families) in an environment of rules called civilization. To us, it matters how fair these rules are, and how they differ from the true wilderness.

The rules of civilization should be better for us than those of wilderness, otherwise failing states and anarchy would be a priority. Believe me, they are not a priority, even if you are an anarchist. If there were anarchy, your standard of living would fall, you would lose many opportunities of entertainment and learning, so you would have less enjoyments, while you would suffer more, you would probably be hungry, you would probably feel cold in winter, etc. Even your life would be in danger. If these were not enough, you might also lose your main reasons of being an anarchist: justice and freedom. Anarchy would not be just, because better equipped and larger groups would probably defeat the smaller ones. After doing so, they might turn you (or the anarchist) to a slave, like the Islamic State did with many people. Moreover, anarchy would not be sustainable either. Sooner or later some kind of military would form, and it would take control and then install some kind of civilization different from anarchy. Humankind would reorganize itself, and organization would conquer the less organized. Thus, failing states and anarchy are not a real choice, they are just temporary. Instead of them, we shall contemplate on different forms of civilization, which one is better than the others...

Well, while anarchists do not offer real solutions, they might still see some problems well. They are good at being against something, and this thing is the unjust distribution of power, including too great power over the poor. In the wilderness, there is no such a huge injustice. Animals only control their own bodies, and even alpha males have to fight for their superiority. This means that probably the ablest individual rules in Nature, if there is any rule at all. In human society, however, this is not necessarily the case... The 20th century has shown us two kinds of societies: capitalism and communism. The anarchists were against both, but they did not manage to introduce their own system, which is not a surprise after what I wrote here. In capitalism, probably the capitalists rule, i. e. those who have a lot of money. Even if there are a lot of opportunities for getting rich (or for getting political power), it is not likely that the ablest (or the wisest, most ethical, most diligent, etc.) individuals rule in capitalism in general. In communism, the political elite rules, probably those who are favoured by former politicians or who knows? The communist systems of the 20th century might not be the same as ideal communism. Ideal communism is democratic, while most of the communist systems of the 20th century were like dictatorships. Anyway, the power of the State is quite strong in communism (as well as the power of one Party), and this power might easily be even worse than the power of capitalists over the poor. Therefore, it is also not likely that the ablest (or the wisest, most ethical, most diligent, etc.) individuals would support a dictatorship or such a strong power of the State (or of „the” Party), so it is not likely that they would generally rule in communism either. (Note that a large state administration is not necessarily bad: but it should be about a „large public sector” instead of the „strong power of the state”.) Thus, we have seen that the political and economic systems of the 20th century were not perfect, and we (or the anarchists) can learn from the wilderness to criticize or improve them.

The point is to improve the rules of civilization, making it almost as just to humans as Nature was to different bears (or maybe even more)… and to make civilization almost as sustainable as Nature was (natural life has a history of millions of years, but civilization has been changing much more rapidly). Thus, in some respects, Nature is the role model, and some of its beneficial functions still missing from civilization should be implemented in society, too. For example, overpopulation should be prevented by laws, instead of letting Nature check it (as the latter would probably bring more suffering and instability). Something like permaculture (or organic farming) should be widespread, instead of letting farmers use genetically modified (GMO) seeds, pesticides and fertilizers, etc. The children of the rich should not inherit a lot of land from their parents (as land was common in Nature). There are many more ideas for a sustainable and equitable society, and these ideas can also be elaborated in more detail and perfection… but the question arises: are you willing to support some kind of law against overpopulation? Are you willing to pay more for better food? Are you willing to vote for big socialist reforms? Yes? Then why don't the politicians speak about these topics more often? Why isn't the limitation of overpopulation among the Sustainable Development Goals of the United Nations? Okay, maybe no one could suggest perfect and detailed laws (maybe „natural laws”) against overpopulation yet, especially not on the global level. In many developed countries, the population is not growing significantly, so the politicians might have thought that this will be the case globally after every country becomes developed… however, population would also grow in developed countries if the prices of houses and of utility bills were cheaper. Some „unassuming” part of their population might still grow. Also, overpopulation should be limited now, not only after every country gets developed, which will either happen or not. Maybe the possible laws against overpopulation would be worse for some countries than for others, and the people at the United Nations did not dare to introduce these restrictions. Anyway, at least I have written a little about these questions in this essay… but let's return to the topic.

Civilization is the new wilderness, and together with the old wilderness they determine the rules of the game of life. Life is like a game, and it does not seem to be a good game if we suffer or lose due to handicaps (but it may still be good if there is reincarnation, Heaven or Nirvana). Democratic Socialism (something between capitalism and communism) may diminish these handicaps. Success in life (and therefore, its enjoyment) may also depend on grace (at least in capitalism): workers can get good jobs by the grace of their employer, and entrepreneurs can sell their goods by the grace of their customers. If someone does not manage to get enough grace, then he or she can still fight or compete. For example, one may try to play chess professionally… Untamed again.

I didn't win on the 33rd Eisaku Sato Essay contest, 2017 (about Demography and the United Nations)

I didn't win on the 33rd Eisaku Sato Essay contest, 2017 (about Demography and the United Nations). Here's the essay I wrote:

Essay Title:
Are all things considered?

Abstract:

As an answer to the theme question of this year, i. e. how the United Nations should cope with recent demographic challenges due to unwished trends, I'm not only giving a consise answer to this question, but also suggest an approach based on philosophy independent from people, places and times, i. e. let it be compatible with the best. It is more interesting to discuss this problem as a domestical issue, so I included that in the essay, and refer to it when writing about the United Nations. Notwithstanding what the majority may vote, Ethics or the Good „God” always has veto power (or freedom not to enforce the laws made by others), and the world is not for the Evil.

Are all things considered?

Where the problems of demography are coming from? Are they coming from a significant difference of genetical traits in people, like some Nazis might have thought, or are they coming from cultural differences that are the result of injustice in the political systems? I think we have no right to judge until we tried to make the political systems more fair (for example, I respect some gypsies for their environmentally friendly low-level activities, but others hate them for similar reasons). Demographic issues are just a subtopic of problems in Political Ethics, and it's not possible to solve them perfectly without solving all the other problems they depend on. This means that a good answer in political ethics should always fit in a plan for a good political system. Injustice will probably exist, however, and it's unethical, as it's hard to be Ethical in all respects, and people often have to use the „principle of lesser evil”. Maybe the Nazis had stopped at this point, and thought: „there will be conflict anyway, and it's easier to win it when we know our men by appearance”. War is not the best solution for the long term, however, as it would quicken harmful technological changes, so it is not part of a good political system, and consequently, it is not the best solution to demographic issues either! As the existence of different animal species is considered good, a xenophobic war could only make things worse, and did not solve anything for the long term, except from survival to that point, maybe, if necessary. The goal should not be confused with the method, and probably even the Nazis could admit that Nazism was a method to invite much more people to the army, almost the entire nation, according to military theories of their times. This was not about philosophical plans for long-term peace (i. e. Political System), but practical improvisation for short-term success. Success, however, is relative to its agent, but the United Nations should support all members equally, by the way. Otherwise, not every people possess the knowledge that is necessary for being Ethical, as Socrates might have said, „Virtue is Knowledge”. This also applies to the knowledge of good political systems, and their reasons. As knowledge increases, the available ethical choices diminish, and some of the former decisions might have to be modified in some degree. For example, the knowledge of good political systems might urge us to make many of our other ethical decisions compatible with the achievement of such a system. This is also true in case of the demographical questions, the actual topic of this essay...

If the United Nations were wrong, or imperfect, then the theme of the 33rd Eisaku Sato Essay Contest could not be perfectly answered either. Thus the discussion of our topic shall be twofold: first discover general truth independent from the actual formation, or actually defining it (as Kant's „League of Nations” idea tried that, for example), then try to apply this in our times to the United Nations, as the actual theme description suggests... so let's begin. In my view, Demographics depends more on economy, and less on culture. This insight comes from my own life: I did not get any wife before I was 34 years old, probably for economic reasons. As it did not seem ethical to strongly pay court to a woman in my position, I did not even venture it, just focused on success and mutual interest on online love finders, which turned to be evil for their licenses, by the way. If I lived alone, my success rates could have been much better. Consequently, I deduced that the size of population is mainly limited by the high costs of living (building, maintenance, overhead expenses) in developed countries. Subsidies and benefits made with the intention of increasing population size do not help, as they usually support those who live below the acceptable standard of living (like many of the gypsies), and like to reproduce even in those circumstances. Economists had long pointed out that this will lead to overpopulation and suffering. That's why the costs of living shall be reduced as a form of subsidy instead of giving blind aid to the people with children. As for the gypsies in extreme poverty, they could afford to live on the same standard of living as common men by the plan mentioned here, and probably they would even do so, and be assimilated into the greater society. This could also help some of the lonely Japanese people to move away from their parents and cure depression thereby (consequently, both the rich and the poor classes could have benefited from this kind of solution). It's possible that without leaving their parents, they cannot find a spouse (or cannot practice business they like). Without finding a good spouse, children are not desired. Thus it can be argued that subsidy on the costs of living would matter more urge towards procreating than subsidy based on the number of children. If this is true, then it is also true that Politicians are responsible for some of the sins that poor people like gypsies do, at least in some degree, which can be greater than 50%. There are not many people who do not accept the opportunities which could mean a win in their lives. This time only the wisest ones would think about the loss of an even greater victory (as a temptation from the Devil, or something like that). As Demography depends on this, it primarily depends on internal affairs or domestic issues, not on international relations. It is possible that low fertility is the revenge of God for injustice in human society (in case it harms sinners more), and injustice is a domestic issue in the first place. Emigration can also be caused by injustice inside the country, when starting a new business is hard. The one-child-policy is also a law of a country (China), not of a greater entity, as this is so strict that makes it unethical to force on political powers or independent communities. Therefore, when searching for good solutions to demographic challenges, we should keep in mind that they are often originated in countries (for their common good), not in diplomatic circles made of agents commissioned for selfish purposes (of one member). In the migrant crisis of 2015-2016, Hungary was such a 'solver' country for obsession (either good or bad) and compulsion. Otherwise it's not sure that politicians will take care of the problems of the world (as problems outside their office) when they could also improve their own countries instead to get the job done well. If the employees of the United Nations were only composed of agents working for countries, the state of the art would not be much better in the field of creative ideas for the common good, I think, and Policy Institutes (also known as: think tanks) could carry more hope than official politicians. They could also try to prove such statements as 'competition due to immigration reduces the growth of the original population, so reducing immigration could increase birth rates'.

For the sake of the common good, it may be really necessary to solve demographic issues by global agreements, to avoid migration problems like we experienced in 2015-2016 CE. This is important as long as there is a risk of war or unwanted migration between countries, and some people might intentionally increase their (male) population size before conquest. Other threat is like the Iron Curtain of the Cold War, not allowing even tourists into other countries by default for mutual distrust, and this way limiting the accessibility of the wonders of the Earth like those called 'World Heritage Sites', as well as famous routes that would prove the maps. In our present world system of countries, therefore, peaceful agreements (like laws) between countries are necessary for the sustainability of the system, and the United Nations organization would make it easier. However, it gets harder again after the United Nations gets alternatives like the G8, G20, the European Union, etc. or maybe even a future organization that will really be better (as a successor of the League of Nations too), and no one knows which one is more important (for example, an Alliance of Small Nations could be a reasonable alternative for those who take it seriously, maybe with unconditional or involuntary membership to make it available to anyone). It's possible that the best solution to the demographic problems fits to an UN-like organization with slightly different rules.

Global agreements shall ideally not force anything, but remain 'agreements'. Otherwise they could build something like the Roman Catholic Church that is hard to change afterwards by 'grassroots' efforts. This is also for ethical philosophy: when something is philosophically evil, and the majority forces the minority to do it, then the majority commits a sin against God, which might lead to some kind of active or passive revenge. In such cases (especially in democratic states with domestic issues) the decision-makers, even more like talk-show hosts, could personally feel the drawbacks of the decision (instead of those they represent or help), and their only protection is that there are too few talented people who wants to be a politician.

Agreements shall not be underestimated, however. Even agreements can be voted democratically, and enforced only by those who voted for it (or even by the others if that's better for them too). Just all agreements should contain some clauses about the things which shall NOT be done with or allowed from those countries which do not enforce the agreement (however, enforcement by activities could kill the organization and therefore, not supported). Embargo, for example, is such a peaceful enforcement. This way it would work in a similar way as unwritten laws: enforcement is done by the community, and in this case, the member countries. When we know that there is nothing else we can ethically do, just some kind of agreement, then thinking time about other solutions can be reduced and the solution made quicker. Now, how could we tackle the demographic problems with this constraint of method?

When it is not ethical to force sending migrants to allied countries as a kind of more equal distribution (as it was planned in the European Union), it could still be ethical to make this a 'liberal' agreement instead of a 'democratically' voted measure, by some kind of 'penalty' or 'alternative choice' for those who do not comply. For example, 'if your country do not accept migrants, ours will send you less money'. There are not two, but more options this way: vote and agree; disagree and exit the organization, or disagree and accept the 'penalty', i. e. that they could deny some new things from us in their land (and the latter also includes the former). The passive penalty (denial, cease) is good because it naturally limits the extent of harm they can cause to member countries, and this way keeps the organization together (it also looks justice, by the way). Moreover, it might also be possible to disagree and vote again after the conditions (of bargain) change.

The United Nations organization should educate politicians and the people about Demography (and other issues) like I tried to do in this essay, but gathering the most beneficial scientific papers with good licences (with endorsement only by employees, not officially), and it could also search for or research solutions like an open policy institute to write the missing ones. When good Science gets weight, more people can relate to it, argue or act accordingly. Public Science is still better than Artifical Intelligence behind the curtains. The United Nations organization should organize (officially initiate) agreements, embargos or similar measures (for discussion and voting) when countries do evil according to the theories in the featured scientific papers (as an office for global common good), and it could recruit more peacekeepers who can be on the spot (not to rule, but to defend or destroy). However, the United Nations should not force anything, as it would endanger the sustainability and the perfection (i. e. justice) of the organization, so it cannot stop wandering migrants alone. In other words, it should have part of the legislative power (some of the reasonable suggestions for discussion) in international law (i. e. voluntary agreements), without any governing (ruling, controlling) power. Unless there are people who work on the global common good officially, it could happen that everyone expects others to work on these problems, but everyone is just doing their own job. Their own job, however, might be in conflict with absolute ethics, as they serve interest groups instead of philosophical rules. As a consequence, phenomena are sometimes not what they seem to be, and this is another reason why the United Nations should not control domestic issues of its members.