# Prime Root Angle Factorization Matrix

The table below demonstrates how the 8 angles vectoring the prime roots in the Prime Spiral Sieve can be used to factorize and account for all composite numbers in the array. There are eight deterministic sequences linked to the prime roots and their angles (reference Croft Spiral Sieve), each possessing a unique initializing and repeating prime root angle multiplier. The second table (bottom of page) shows the repeating patterns for the prime root angle multipliers.

 Prime Roots (pr) 1 7 11 13 17 19 23 29 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 0.5 = A 6 42 66 78 102 114 138 174 A + pr = B 7 49 77 91 119 133 161 203 Factor: B/pr x pr 7 x 1 7 x 7 7 x 11 7 x 13 7 x 17 7 x 19 7 x 23 7 x 29 Prime Roots (pr) + 30 31 37 41 43 47 49 53 59 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 0.5 x (pr + 30) = A 186 222 246 258 282 294 318 354 A + (pr + 30) = B 217 259 287 301 329 343 371 413 Factor: B/pr x pr 7 x 31 7 x 37 7 x 41 7 x 43 7 x 47 7 x 49 7 x 53 7 x 59 ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n}

 Prime Roots (pr) 1 7 11 13 17 19 23 29 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 1.0 = A 12 84 132 156 204 228 276 348 A - pr = B 11 77 121 143 187 209 253 319 Factor: B/pr x pr 11 x 1 11 x 7 11 x 11 11 x 13 11 x 17 11 x 19 11 x 23 11 x 29 Prime Roots (pr) + 30 31 37 41 43 47 49 53 59 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 1.0 x (pr + 30) = A 372 444 492 516 564 588 636 708 A - (pr + 30) = B 341 407 451 473 517 539 583 649 Factor: B/pr x pr 11 x 31 11 x 37 11 x 41 11 x 43 11 x 47 11 x 49 11 x 53 11 x 59 ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n}

 Prime Roots (pr) 1 7 11 13 17 19 23 29 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 1.0 = A 12 84 132 156 204 228 276 348 A + pr = B 13 91 143 169 221 247 299 377 Factor: B/pr x pr 13 x 1 13 x 7 13 x 11 13 x 13 13 x 17 13 x 19 13 x 23 13 x 29 Prime Roots (pr) + 30 31 37 41 43 47 49 53 59 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 1.0 x (pr + 30) = A 372 444 492 516 564 588 636 708 A + (pr + 30) = B 403 481 533 559 611 637 689 767 Factor: B/pr x pr 13 x 31 13 x 37 13 x 41 13 x 43 13 x 47 13 x 49 13 x 53 13 x 59 ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n}

 Prime Roots (pr) 1 7 11 13 17 19 23 29 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 1.5 = A 18 126 198 234 306 342 414 522 A - pr = B 17 119 187 221 289 323 391 493 Factor: B/pr x pr 17 x 1 17 x 7 17 x 11 17 x 13 17 x 17 17 x 19 17 x 23 17 x 29 Prime Roots (pr) + 30 31 37 41 43 47 49 53 59 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 1.5 x (pr + 30) = A 558 666 738 774 846 882 954 1062 A - (pr + 30) = B 527 629 697 731 799 833 901 1003 Factor: B/pr x pr 17 x 31 17 x 37 17 x 41 17 x 43 17 x 47 17 x 49 17 x 53 17 x 59 ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n}

 Prime Roots (pr) 1 7 11 13 17 19 23 29 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 1.5 = A 18 126 198 234 306 342 414 522 A + pr = B 19 133 209 247 323 361 437 551 Factor: B/pr x pr 19 x 1 19 x 7 19 x 11 19 x 13 19 x 17 19 x 19 19 x 23 19 x 29 Prime Roots (pr) + 30 31 37 41 43 47 49 53 59 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 1.5 x (pr + 30) = A 558 666 738 774 846 882 954 1062 A + (pr + 30) = B 589 703 779 817 893 931 1007 1121 Factor: B/pr x pr 19 x 31 19 x 37 19 x 41 19 x 43 19 x 47 19 x 49 19 x 53 19 x 59 ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n}

 Prime Roots (pr) 1 7 11 13 17 19 23 29 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 2.0 = A 24 168 264 312 408 456 552 696 A - pr = B 23 161 253 299 391 437 529 667 Factor: B/pr x pr 23 x 1 23 x 7 23 x 11 23 x 13 23 x 17 23 x 19 23 x 23 23 x 29 Prime Roots (pr) + 30 31 37 41 43 47 49 53 59 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 2.0 x (pr + 30) = A 744 888 984 1032 1128 1176 1272 1416 A - (pr + 30) = B 713 851 943 989 1081 1127 1219 1357 Factor: B/pr x pr 23 x 31 23 x 37 23 x 41 23 x 43 23 x 47 23 x 49 23 x 53 23 x 59 ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n}

 Prime Roots (pr) 1 7 11 13 17 19 23 29 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 2.5 = A 30 210 330 390 510 570 690 870 A - pr = B 29 203 319 377 493 551 667 841 Factor: B/pr x pr 29 x 1 29 x 7 29 x 11 29 x 13 29 x 17 29 x 19 29 x 23 29 x 29 Prime Roots (pr) + 30 31 37 41 43 47 49 53 59 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 2.5 x (pr + 30) = A 930 1110 1230 1290 1410 1470 1590 1770 A - (pr + 30) = B 899 1073 1189 1247 1363 1421 1537 1711 Factor: B/pr x pr 29 x 31 29 x 37 29 x 41 29 x 43 29 x 47 29 x 49 29 x 53 29 x 59 ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n}

 Prime Roots (pr) 1 7 11 13 17 19 23 29 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 2.5 = A 30 210 330 390 510 570 690 870 A + pr = B 31 217 341 403 527 589 713 899 Factor: B/pr x pr 31 x 1 31 x 7 31 x 11 31 x 13 31 x 17 31 x 19 29 x 23 31 x 29 Prime Roots (pr) + 30 31 37 41 43 47 49 53 59 Prime Root Angles (pr∠) 12° 84° 132° 156° 204° 228° 276° 348° pr∠ x 2.5 x (pr + 30) = A 930 1110 1230 1290 1410 1470 1590 1770 A + (pr + 30) = B 961 1147 1271 1333 1457 1519 1643 1829 Factor: B/pr x pr 31 x 31 31 x 37 31 x 41 31 x 43 31 x 47 31 x 49 31 x 53 31 x 59 ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n} ⇓ {+ 30...n}

## Prime Root Angle Multiplier Sequence

 Multiplier + or - Prime Root Initial Root Factors ... n 0.0 + 1 x 1 ... n 0.5 + 7 x 1 ... n 1.0 - 11 x 1 ... n 1.0 + 13 x 1 ... n 1.5 - 17 x 1 ... n 1.5 + 19 x 1 ... n 2.0 - 23 x 1 ... n 2.5 - 29 x 1 ... n 2.5 + 31 x 1 ... n 3.0 + 37 x 1 ... n 3.5 - 41 x 1 ... n 3.5 + 43 x 1 ... n 4.0 - 47 x 1 ... n 4.0 + 49 x 1 ... n 4.5 - 53 x 1 ... n 5.0 - 59 x 1 ... n 5.0 + 61 x 1 ... n 5.5 + 67 x 1 ... n 6.0 - 71 x 1 ... n 6.0 + 73 x 1 ... n 6.5 - 77 x 1 ... n 6.5 + 79 x 1 ... n 7.0 - 83 x 1 ... n 7.5 - 89 x 1 ... n 7.5 + 91 x 1 ... n 8.0 + 97 x 1 ... n 8.5 - 101 x 1 ... n 8.5 + 103 x 1 ... n 9.0 - 107 x 1 ... n 9.0 + 109 x 1 ... n 9.5 - 113 x 1 ... n 10.0 - 119 x 1 ... n