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Prime Root Angle Factorization Matrix

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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