*nobody*really knows a full explanation of.

Like, I accepted long ago that the distribution of primes themselves was sort of random, even as abhorrent as it feels to accept that some random bullshit arises out of such a simple definition (and it's not even

*pretty*random bullshit in the same sense as the mandelbrot set arising out of just z

^{2}+ c). But the distribution of which numbers are primitive roots in

**Z**/(p)?

Just looks like a bunch of garbage, with the exception of vertical white stripes because perfect squares can't be primitive roots. X-axis is natural numbers n, Y-axis is primes p, and the coloration is black if n is a "primitive root mod p", that is, if the powers 1,n,n^2,...n^(p-2) are all distinct, white if not, and gray if n is too big to even be eligible. And then GF(p^n) for n≥1 are even

*more*crazy. What the fuck, number theory.

#!/usr/bin/perl @primes = qw( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 ); sub order { my ($p, $m) = @_; return 0 if $m < 1; return 0 if $m >= $p; my $x = 1; my $t = 0; while (1) { $t++; # print "$x "; $x = ($x * $m) % $p; return $t if $x == 1; } } print " 2 3 4 5 6 7 8 9 " . ("0 1 2 3 4 5 6 7 8 9 " x 2); print "\n"; for my $p (@primes) { # for my $m (1..$p-1) { printf "% 5d ", $p; for my $k (2..29) { my $o = order($p, $k); print $o == 0 ? " " : $o == $p-1 ? "# " : "' "; } print "\n"; }

#!/usr/bin/perl @primes = qw( 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 ); sub order { my ($p, $m) = @_; return 0 if $m < 1; return 0 if $m >= $p; my $x = 1; my $t = 0; while (1) { $t++; # print "$x "; $x = ($x * $m) % $p; return $t if $x == 1; } } my $w = 240; my $h = @primes; print "P2 $w $h 255\n"; for my $p (@primes) { for my $k (2..$w+1) { my $o = order($p, $k); print $o == 0 ? "128 " : $o == $p-1 ? "0 " : "255 "; } print "\n"; }