1.7.4 Primality

The prime num­bers are dis­tin­guished from the non-primes by their common co­or­di­nates at the in­ter­sec­tion of the $g_{p}$ and $e^{\left( \frac{2i\pi}{n}\left\lbrack (n - 1)! \right\rbrack \right)}$ curves. In­tu­itive­ly, the fac­to­r­i­al func­tion as $(n - 1)!$rep­re­sents the means to check for even di­vis­i­bil­i­ty for a nat­ur­al number $n$. Taking the $(n - 1)!$fac­to­r­i­al, we can create an $(n - 1) \times (n - 1)$ matrix array where each row/column entry $(j,k)$ equals

which is im­ple­ment­ed using the fol­low­ing MATLAB code for a spec­i­fied nat­ur­al number $n:$

NumM= [];

n1=n;

for j=1:(n1-1)

vec= [];

for k=1:(n1-1)

iter=mod(j*k, n1);

vec= [vec, iter];

end

NumM= [NumM; vec];

end

We use this al­go­rithm to eval­u­ate the nat­ur­al num­bers from $5$ to $10$, ar­rang­ing these ma­tri­ces in the array

1	2	3	4	5	6	7	8	9
$\textcolor[HTML]{0070C0}{2}$	$\textcolor[HTML]{0070C0}{4}$	$\textcolor[HTML]{0070C0}{6}$	$\textcolor[HTML]{0070C0}{8}$	$\textcolor[HTML]{FF0000}{0}$	$\textcolor[HTML]{0070C0}{2}$	$\textcolor[HTML]{0070C0}{4}$	$\textcolor[HTML]{0070C0}{6}$	$\textcolor[HTML]{0070C0}{8}$
$\textcolor[HTML]{0070C0}{3}$	$\textcolor[HTML]{0070C0}{6}$	$\textcolor[HTML]{0070C0}{9}$	$\textcolor[HTML]{0070C0}{2}$	$\textcolor[HTML]{0070C0}{5}$	$\textcolor[HTML]{0070C0}{8}$	$\textcolor[HTML]{0070C0}{1}$	$\textcolor[HTML]{0070C0}{4}$	$\textcolor[HTML]{0070C0}{7}$
4	8	2	6	$\textcolor[HTML]{FF0000}{0}$	4	8	2	6
$\textcolor[HTML]{0070C0}{5}$	$\textcolor[HTML]{FF0000}{0}$	$\textcolor[HTML]{0070C0}{5}$	$\textcolor[HTML]{FF0000}{0}$	$\textcolor[HTML]{0070C0}{5}$	$\textcolor[HTML]{FF0000}{0}$	$\textcolor[HTML]{0070C0}{5}$	$\textcolor[HTML]{FF0000}{0}$	$\textcolor[HTML]{0070C0}{5}$
6	2	8	4	$\textcolor[HTML]{FF0000}{0}$	6	2	8	4
$\textcolor[HTML]{0070C0}{7}$	$\textcolor[HTML]{0070C0}{4}$	$\textcolor[HTML]{0070C0}{1}$	$\textcolor[HTML]{0070C0}{8}$	$\textcolor[HTML]{0070C0}{5}$	$\textcolor[HTML]{0070C0}{2}$	$\textcolor[HTML]{0070C0}{9}$	$\textcolor[HTML]{0070C0}{6}$	$\textcolor[HTML]{0070C0}{3}$
8	6	4	2	$\textcolor[HTML]{FF0000}{0}$	8	6	4	2
9	8	7	6	5	4	3	2	1

The $M_{n}$ ma­tri­ces are dis­played in Table 1.7.4, which are sym­met­ric, and each equals its own matrix trans­pose. The in­ter­pre­ta­tion of the po­si­tion of the zeros in the com­pos­ite num­bers is the row/column num­bers rep­re­sent the spe­cif­ic evenly di­vis­i­ble prod­uct(s). For ex­am­ple, in the $M_{9}$ matrix, the zeros co­in­cide with prod­ucts gen­er­at­ed by $(3,3),(3,6),(6,3),(6,6)$, re­call­ing this is con­gru­ent modulo 9. In each of the $M_{n}$ ma­tri­ces sub­ma­tri­ces are high­light­ed in grey and light blue, with the orange cross­es rep­re­sent­ing shared rows/columns in the sub­ma­tri­ces for even-valued $n$. We draw at­ten­tion to the odd-valued ma­tri­ces in par­tic­u­lar and the di­ag­o­nal and anti-di­ag­o­nal of these sub­ma­tri­ces, which we eval­u­ate con­gru­ent modulo $n$ the prod­uct of the el­e­ments in $M_{5},M_{7},M_{9}$

These con­gru­ent modulo di­ag­o­nal/anti-di­ag­o­nal el­e­ment prod­ucts yield values con­sis­tent with the pri­ma­ry gen­er­a­tors ori­ent­ed coun­ter­clock­wise and clock­wise from the iden­ti­ty for the prime num­bers.

By the in­her­ent com­plex geom­e­try, pri­mal­i­ty test­ing can be achieved by en­sur­ing each el­e­ment $\in G_{n}$ trans­forms by a power to equal one of the two guar­an­teed gen­er­a­tors. Using the ex­po­nents in the Geo­met­ric Num­bers, a matrix array can be cre­at­ed as fol­lows for $G_{7}$

where each el­e­ment in the matrix por­tion above rep­re­sents each vertex el­e­ment from the $7$-gon raised to the powers $1$ through $7\ mod(7).$The guar­an­teed gen­er­a­tors $\in G_{7}$ are $e^{2i\pi\left( \frac{1}{7} \right)}$ and $e^{2i\pi\left( \frac{6}{7} \right)}$, which by powers, will form the entire group

The zeros in row $7$ and column $7$ rep­re­sent the iden­ti­ty and are triv­ial in these po­si­tions for pri­mal­i­ty test­ing. We remove the outer ring of se­quen­tial num­bers formed by $\left\langle e^{2i\pi\left( \frac{1}{7} \right)} \right\rangle,\left\langle e^{2i\pi\left( \frac{6}{7} \right)} \right\rangle$, then high­light in red the values that equal the gen­er­a­tors $\left\langle e^{2i\pi\left( \frac{1}{7} \right)} \right\rangle,\left\langle e^{2i\pi\left( \frac{6}{7} \right)} \right\rangle$ in the ex­pres­sion below

find­ing that a sym­met­ric ring of gen­er­a­tors forms in the matrix. The gen­er­a­tor power hits ensure the four other non-iden­ti­ty el­e­ments in $G_{7}$ are also gen­er­a­tors of the group, show­ing that $G_{7}\in \mathbb{P}.$To clear­ly see this, the non-iden­ti­ty ver­tices $\in G_{7}$ in ex­po­nen­tial form when raised to se­quen­tial powers have the rep­re­sen­ta­tion

Ex­press­ing the $4 \times 4$ sub­ma­trix of powers $\in G_{7}$ in this form, we find

which sim­pli­fies after cal­cu­lat­ing the power prod­ucts and per­form­ing modulo arith­metic to

where once again the gen­er­a­tors $\left\langle e^{2i\pi\left( \frac{1}{7} \right)} \right\rangle,\left\langle e^{2i\pi\left( \frac{6}{7} \right)} \right\rangle$ are high­light­ed in red. The first power test for $e^{\left( \frac{2i\pi}{7}\lbrack j \bullet k\ \ \ mod\ 7\rbrack \right)}$ that lands on a gen­er­a­tor is suf­fi­cient to show that $e^{\left( \frac{2i\pi}{7}\lbrack k\rbrack \right)}$ is also a gen­er­a­tor of $G_{7}.$

Theory (1.7.383) – For $G_{n}$ a finite Geo­met­ric Number $\in \mathbb{C}$, the com­plex con­ju­gate of any gen­er­a­tor $\left\langle e^{\left( \frac{2i\pi}{n}\lbrack k\rbrack \right)} \right\rangle = G_{n}$ is also a gen­er­a­tor of the group $\left\langle e^{\left( \frac{2i\pi}{n}\lbrack k\rbrack \right)} \right\rangle = G_{n}.$

Proof $\mathbf{-}$ The com­plex con­ju­gate for any non-iden­ti­ty vertex $\in G_{n}$, with vertex number $k\in \mathbb{N}$ ori­ent­ed on the $n$-gon, equals

By the sym­me­try of $U(1)$, both ver­tices are an equal dis­tance in ra­di­ans on the bound­ary of the com­plex unit disc from the iden­ti­ty at $e^{0} = \epsilon.$ Given that $e^{2i\pi\left( \frac{k}{n} \right)}$ is a gen­er­a­tor, it lands on either $e^{2i\pi\left( \frac{1}{n} \right)}$ and $e^{2i\pi\left( \frac{(n - 1)}{n} \right)}$ for a given power $j\in \mathbb{N}$ $\left( e^{2i\pi\left( \frac{j \bullet k}{n} \right)} \right)$. The equal radian dis­tance that $e^{- 2i\pi\left( \frac{j \bullet k}{n} \right)}$ trav­els then lands on the con­ju­gate gen­er­a­tor to that of $e^{2i\pi\left( \frac{j \bullet k}{n} \right)}$, and is, there­fore, a gen­er­a­tor $\in G_{n}.$

$\square$

From the pre­vi­ous matrix, the red vertex en­tries in the upper half (first two rows) are mir­rored in the lower half by their com­plex con­ju­gates, given that

The same con­ju­gate re­la­tion­ship holds for the left and right two columns. This im­plies that for the $G_{7}$, gen­er­a­tor, test­ing is only re­quired for the two ver­tices coun­ter­clock­wise from $e^{2i\pi\left( \frac{1}{7} \right)}$, or the two ver­tices clock­wise from $e^{2i\pi\left( \frac{6}{7} \right)}.$In layper­son's terms, check­ing the non-iden­ti­ty ver­tices in the top or bottom half is suf­fi­cient to test pri­mal­i­ty.

Wilson's the­o­rem states that prime num­bers obey the fol­low­ing

The the­o­rem can be ap­plied by re­turn­ing to the $G_{7}$ power table, where color in blue the sub­ma­tri­ces with values as­so­ci­at­ed with the $e^{2i\pi\left( \frac{6}{7} \right)}$ gen­er­a­tor, and in green, the $e^{2i\pi\left( \frac{1}{7} \right)}$ gen­er­a­tor.

We cal­cu­late the prod­uct of the el­e­ments lo­cat­ed on the anti-di­ag­o­nal modulo $7$

which is con­sis­tent with the the­o­rem, given that for $p = 7$, we find

and $(7 - 1) \equiv 6$. In the Geo­met­ric Num­bers the value $k = 6$ rep­re­sents the $e^{2i\pi\left( \frac{6}{7} \right)}$ gen­er­a­tor. By Fermat's little the­o­rem

we only need to cal­cu­late the anti-di­ag­o­nal from one green sub­ma­trix, given that $2 \in \mathbb{P}$, find­ing the same result

This case's values formed the anti-di­ag­o­nal from the sub­ma­tri­ces con­tain­ing the $e^{2i\pi\left( \frac{1}{7} \right)}$ gen­er­a­tor. To con­trast, we can per­form the same cal­cu­la­tion using the anti-di­ag­o­nal from a blue sub­ma­trix, find­ing

which rep­re­sents the other gen­er­a­tor $e^{2i\pi\left( \frac{1}{7} \right)}$. For the reader, we in­clude the matrix array rep­re­sen­ta­tions for three larger nat­ur­al num­bers in the fol­low­ing tables, one non-prime $n = 15$, and the twin prime pair $(n,n + 2) = (17,19)$, which rep­re­sent the Geo­met­ric Num­bers $G_{15},G_{17},$and $G_{19}$. Each array has color-coded sub­ma­tri­ces that rep­re­sent pair­ings of trans­pose sym­me­try in the array for each number.

1	2	3	4	5	6	7	8	9	10	11	12	13	14
2	4	6	8	10	12	$\textcolor[HTML]{FF0000}{14}$	$\textcolor[HTML]{FF0000}{1}$	3	5	7	9	11	13
3	6	9	12	$\textcolor[HTML]{C45911}{0}$	3	6	9	12	$\textcolor[HTML]{C45911}{0}$	3	6	9	12
4	8	12	$\textcolor[HTML]{FF0000}{1}$	5	9	13	2	6	10	$\textcolor[HTML]{FF0000}{14}$	3	7	11
5	10	$\textcolor[HTML]{C45911}{0}$	5	10	$\textcolor[HTML]{C45911}{0}$	5	10	$\textcolor[HTML]{C45911}{0}$	5	10	$\textcolor[HTML]{C45911}{0}$	5	10
6	12	3	9	$\textcolor[HTML]{C45911}{0}$	6	12	3	9	$\textcolor[HTML]{C45911}{0}$	6	12	3	9
7	$\textcolor[HTML]{FF0000}{14}$	6	13	5	12	4	11	3	10	2	9	$\textcolor[HTML]{FF0000}{1}$	8
8	$\textcolor[HTML]{FF0000}{1}$	9	2	10	3	11	4	12	5	13	6	$\textcolor[HTML]{FF0000}{14}$	7
9	3	12	6	$\textcolor[HTML]{C45911}{0}$	9	3	12	6	$\textcolor[HTML]{C45911}{0}$	9	3	12	6
10	5	$\textcolor[HTML]{C45911}{0}$	10	5	$\textcolor[HTML]{C45911}{0}$	10	5	$\textcolor[HTML]{C45911}{0}$	10	5	$\textcolor[HTML]{C45911}{0}$	10	5
11	7	3	$\textcolor[HTML]{FF0000}{14}$	10	6	2	13	9	5	$\textcolor[HTML]{FF0000}{1}$	12	8	4
12	9	6	3	$\textcolor[HTML]{C45911}{0}$	12	9	6	3	$\textcolor[HTML]{C45911}{0}$	12	9	6	3
13	11	9	7	5	3	$\textcolor[HTML]{FF0000}{1}$	$\textcolor[HTML]{FF0000}{14}$	12	10	8	6	4	2
14	13	12	11	10	9	8	7	6	5	4	3	2	1

As ex­pect­ed, the non-prime $G_{15}$ has a total of six­teen zeros in its matrix array rep­re­sen­ta­tion, the even di­vis­i­bil­i­ty of its prime de­com­po­si­tion num­bers $p_{2},p_{3} = 3,5.$Eight pos­si­ble trans­pose sub-sym­me­tries in $G_{15}$, are seen by the match­ing color sub­ma­trix pairs. Taken as a whole, a global sym­me­try exists as two trans­pose sym­me­tries, rep­re­sent­ed by the amal­gam of all blue-shaded and green-shaded sub­ma­tri­ces.

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
2	4	6	8	10	12	14	$\textcolor[HTML]{FF0000}{16}$	$\textcolor[HTML]{FF0000}{1}$	3	5	7	9	11	13	15
3	6	9	12	15	$\textcolor[HTML]{FF0000}{1}$	4	7	10	13	$\textcolor[HTML]{FF0000}{16}$	2	5	8	11	14
4	8	12	$\textcolor[HTML]{FF0000}{16}$	3	7	11	15	2	6	10	14	$\textcolor[HTML]{FF0000}{1}$	5	9	13
5	10	15	3	8	13	$\textcolor[HTML]{FF0000}{1}$	6	11	$\textcolor[HTML]{FF0000}{16}$	4	9	14	2	7	12
6	12	$\textcolor[HTML]{FF0000}{1}$	7	13	2	8	14	3	9	15	4	10	$\textcolor[HTML]{FF0000}{16}$	5	11
7	14	4	11	$\textcolor[HTML]{FF0000}{1}$	8	15	5	12	2	9	$\textcolor[HTML]{FF0000}{16}$	6	13	3	10
8	$\textcolor[HTML]{FF0000}{16}$	7	15	6	14	5	13	4	12	3	11	2	10	$\textcolor[HTML]{FF0000}{1}$	9
9	$\textcolor[HTML]{FF0000}{1}$	10	2	11	3	12	4	13	5	14	6	15	7	$\textcolor[HTML]{FF0000}{16}$	8
10	3	13	6	$\textcolor[HTML]{FF0000}{16}$	9	2	12	5	15	8	$\textcolor[HTML]{FF0000}{1}$	11	4	14	7
11	5	$\textcolor[HTML]{FF0000}{16}$	10	4	15	9	3	14	8	2	13	7	$\textcolor[HTML]{FF0000}{1}$	12	6
12	7	2	14	9	4	$\textcolor[HTML]{FF0000}{16}$	11	6	$\textcolor[HTML]{FF0000}{1}$	13	8	3	15	10	5
13	9	5	$\textcolor[HTML]{FF0000}{1}$	14	10	6	2	15	11	7	3	16	12	8	4
14	11	8	5	2	$\textcolor[HTML]{FF0000}{16}$	13	10	7	4	$\textcolor[HTML]{FF0000}{1}$	15	12	9	6	3
15	13	11	9	7	5	3	$\textcolor[HTML]{FF0000}{1}$	$\textcolor[HTML]{FF0000}{16}$	14	12	10	8	6	4	2
16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1

In Table 1.7.6 and Table 1.7.7, all en­tries in each array are non-zero, which is con­sis­tent with the non-di­vis­i­bil­i­ty of the prime num­bers. There are only two trans­form sym­me­tries for $G_{17}$ and $G_{19}$, con­sis­tent with one global sym­me­try for their power prod­ucts.

1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
2	4	6	8	10	12	14	16	$\textcolor[HTML]{FF0000}{18}$	$\textcolor[HTML]{FF0000}{1}$	3	5	7	9	11	13	15	17
3	6	9	12	15	$\textcolor[HTML]{FF0000}{18}$	2	5	8	11	14	17	$\textcolor[HTML]{FF0000}{1}$	4	7	10	13	16
4	8	12	16	$\textcolor[HTML]{FF0000}{1}$	5	9	13	17	2	6	10	14	$\textcolor[HTML]{FF0000}{18}$	3	7	11	15
5	10	15	$\textcolor[HTML]{FF0000}{1}$	6	11	16	2	7	12	17	3	8	13	$\textcolor[HTML]{FF0000}{18}$	4	9	14
6	12	$\textcolor[HTML]{FF0000}{18}$	5	11	17	4	10	16	3	9	15	2	8	14	$\textcolor[HTML]{FF0000}{1}$	7	13
7	14	2	9	16	4	11	$\textcolor[HTML]{FF0000}{18}$	6	13	$\textcolor[HTML]{FF0000}{1}$	8	15	3	10	17	5	12
8	16	5	13	2	10	$\textcolor[HTML]{FF0000}{18}$	7	15	4	12	$\textcolor[HTML]{FF0000}{1}$	9	17	6	14	3	11
9	$\textcolor[HTML]{FF0000}{18}$	8	17	7	16	6	15	5	14	4	13	3	12	2	11	$\textcolor[HTML]{FF0000}{1}$	10
10	$\textcolor[HTML]{FF0000}{1}$	11	2	12	3	13	4	14	5	15	6	16	7	17	8	$\textcolor[HTML]{FF0000}{18}$	9
11	3	14	6	17	9	$\textcolor[HTML]{FF0000}{1}$	12	4	15	7	$\textcolor[HTML]{FF0000}{18}$	10	2	13	5	16	8
12	5	17	10	3	15	8	$\textcolor[HTML]{FF0000}{1}$	13	6	$\textcolor[HTML]{FF0000}{18}$	11	4	16	9	2	14	7
13	7	$\textcolor[HTML]{FF0000}{1}$	14	8	2	15	9	3	16	10	4	17	11	5	$\textcolor[HTML]{FF0000}{18}$	12	6
14	9	4	$\textcolor[HTML]{FF0000}{18}$	13	8	3	17	12	7	2	16	11	6	$\textcolor[HTML]{FF0000}{1}$	15	10	5
15	11	7	3	$\textcolor[HTML]{FF0000}{18}$	14	10	6	2	17	13	9	5	$\textcolor[HTML]{FF0000}{1}$	16	12	8	4
16	13	10	7	4	$\textcolor[HTML]{FF0000}{1}$	17	14	11	8	5	2	$\textcolor[HTML]{FF0000}{18}$	15	12	9	6	3
17	15	13	11	9	7	5	3	$\textcolor[HTML]{FF0000}{1}$	$\textcolor[HTML]{FF0000}{18}$	16	14	12	10	8	6	4	2
18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1

There is a pro­found beauty when the prime num­bers are seen in the com­plex geom­e­try in­stead of the nat­ur­al num­bers on the line. To reveal this beauty, we define the pos­i­tive in­te­gers by the coun­ter­clock­wise ori­ent­ed guar­an­teed pri­ma­ry gen­er­a­tors for each geo­met­ric $n$-gon using the fol­low­ing para­met­ric wave func­tion

Specif­i­cal­ly, the $g^{+}\left( \mathbb{Z}^{+}(n) \right)$ wave­form in­ter­sects each $n$-gon at the cor­re­spond­ing pri­ma­ry gen­er­a­tor vertex $g_{n}^{+}\left( e^{2i\pi\left\lbrack \frac{1}{n} \right\rbrack} \right)$. The $g^{+}\left( \mathbb{Z}^{+}(n) \right)$ wave­form is plot­ted in Figure 1.7.36 for prime num­bers $g_{p}\in \mathbb{P}$ in green, and non-prime num­bers $g_{n}\notin \mathbb{P}$ in blue. For $n \geq 3$, chords con­nect the pos­i­tive and neg­a­tive guar­an­teed pri­ma­ry gen­er­a­tor ver­tices.

The $g^{+}\left( \mathbb{Z}^{+}(n) \right)$ wave­form can be ro­tat­ed around the $x$-axis to by the spe­cif­ic ar­gu­ment value $2i\pi\left\lbrack \frac{1}{n} \right\rbrack$, where $n$ rep­re­sents the in­te­ger value of the nat­ur­al num­bers $g_{n}$. We chose to rotate $g^{+}\left( \mathbb{Z}^{+}(n) \right)$ in the neg­a­tive/clock­wise ori­en­ta­tion, man­i­fest­ed by the fol­low­ing para­met­ric func­tion

Geo­met­ric Primes $\left\langle g_{p} \right\rangle\in \mathbb{P}$ are la­belled for the first eight prime num­bers and $\left\langle g_{1} \right\rangle$.

For the in­di­vid­ual nat­ur­al number rep­re­sen­ta­tions of $\mathbb{N}\left( g_{n} \right)$, the wave­form will in­ter­sect the spe­cif­ic $n\in \mathbb{N}$ at the iden­ti­ty el­e­ment, shown in Figure 1.7.37 for the first eight prime num­bers. For each $g_{n}$, the $n$-gon rep­re­sen­ta­tion is con­veyed by the points on the unit cir­cles for each of its $n$ ver­tices.

Evenly di­vis­i­bil­i­ty can be seen geo­met­ri­cal­ly for $g_{n}\notin \mathbb{P}$ by wave­form in­ter­sects for these num­bers by the $\mathbb{P}\left( g_{p} \right)$ wave­forms that com­prise the said num­bers prime de­com­po­si­tion. For ex­am­ple, the $g_{3}$ wave­form can be seen to in­ter­sec­tion ver­tices of the blue unit cir­cles at $x = \left\{ g_{6},g_{9},g_{12},g_{15},g_{18} \right\}$. The $\mathbb{N}\left( g_{n} \right)$ in­ter­sec­tions with the iden­ti­ty el­e­ment of each $g_{n}$ is seen in Figure 1.7.38 at the pos­i­tive peaks along the $y$-axis for each wave­form.

We define the pos­i­tive in­te­ger func­tion $g^{-}\left( \mathbb{Z}^{+}(n) \right)$, as the wave­form in­ter­sect­ing with each neg­a­tive ori­ent­ed guar­an­teed pri­ma­ry gen­er­a­tor $\forall\ g_{n}$

which dif­fers from $g^{+}\left( \mathbb{Z}^{+}(n) \right)$ by the sign for the imag­i­nary por­tion of the para­met­ric func­tion. The $g^{+}\left( \mathbb{Z}^{+}(n) \right)$ and $g^{-}\left( \mathbb{Z}^{+}(n) \right)$ wave func­tions are plot­ted in Figure 1.7.39, in which, it is straight­for­ward to see the wave­forms in­ter­sect with the ver­tices of each $n$-gon in a com­plex con­ju­gate re­la­tion­ship to one an­oth­er.