1.7.3 Prime Number Theory and the Geometry of Groups

Wilson's the­o­rem rep­re­sents es­sen­tial ma­te­r­i­al in any in­tro­duc­to­ry prime number theory course. The the­o­rem is named after John Wilson¹, an 18th-cen­tu­ry math­e­mati­cian, al­though it had been stated in the 11^th cen­tu­ry by the Per­sian math­e­mati­cian Ibn al-Haytham². Wilson's the­o­rem states that a nat­ur­al number $n$ is prime if and only if

Wilson's the­o­rem is for­mal­ly stated by the fol­low­ing:

The­o­rem (1.7.272) – Wilson's The­o­rem: A nat­ur­al number $p$ is prime $\left( p\in \mathbb{P} \right)$ if and only if the prod­uct of all the pos­i­tive in­te­gers up to $p - 1$, or $(p - 1)!$, is one less than a mul­ti­ple of $p$

Using the prime number ex­am­ple $p = 5$, we find

which is one less than $5$, a mul­ti­ple of $p = 5.$By ma­nip­u­la­tion, the the­o­rem can be ex­pressed as ini­tial­ly stated

and our $p = 5$ ex­am­ple can be seen to hold

For a non-prime nat­ur­al number $n \notin P$, we choose $n = 6$

find­ing the result is equal to $1$, and in the stated the­o­rem's form, we would find it equals zero

In se­quence no­ta­tion, Wilson's the­o­rem can be ex­pressed as

which gen­er­ates the fol­low­ing table of values for $n = \lbrack 1,100\rbrack$.

$\displaystyle n = (k + j)$	$\displaystyle j = 1$	$\displaystyle j = 2$	$\displaystyle j = 3$	$\displaystyle j = 4$	$\displaystyle j = 5$	$\displaystyle j = 6$	$\displaystyle j = 7$	$\displaystyle j = 8$	$\displaystyle j = 9$	$\displaystyle j = 10$
$\displaystyle k = 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 1}$	$\textcolor[HTML]{FF0000}{\displaystyle 2}$	$\displaystyle 2$	$\textcolor[HTML]{FF0000}{\displaystyle 4}$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 6}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$
$\displaystyle k = 10$	$\textcolor[HTML]{FF0000}{\displaystyle 10}$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 12}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 16}$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 18}$	$\displaystyle 0$
$\displaystyle k = 20$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 22}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 28}$	$\displaystyle 0$
$\displaystyle k = 30$	$\textcolor[HTML]{FF0000}{\displaystyle 30}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 36}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$
$\displaystyle k = 40$	$\textcolor[HTML]{FF0000}{\displaystyle 40}$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 42}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 46}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$
$\displaystyle k = 50$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 52}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 58}$	$\displaystyle 0$
$\displaystyle k = 60$	$\textcolor[HTML]{FF0000}{\displaystyle 60}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	0	$\textcolor[HTML]{FF0000}{\displaystyle 66}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$
$\displaystyle k = 70$	$\textcolor[HTML]{FF0000}{\displaystyle 70}$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 72}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 78}$	$\displaystyle 0$
$\displaystyle k = 80$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 82}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 88}$	$\displaystyle 0$
$\displaystyle k = 90$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$	$\textcolor[HTML]{FF0000}{\displaystyle 96}$	$\displaystyle 0$	$\displaystyle 0$	$\displaystyle 0$

In the table above, all non-zero num­bers are prime and in red, except for $n = (0 + 4) = 4$

which is the only non-prime nat­ur­al number equal to a non-zero result. The the­o­rem still holds, given that $2 + 1$ is not evenly di­vis­i­ble by $4$. We remark that the prime num­bers appear to ag­gre­gate in the fol­low­ing set of odd-valued columns $j = \left\{ 1,3,7,9 \right\}$. The rea­son­ing should be easy to see for the even-valued columns, given that $n = 2$ will divide all the even num­bers within. For the odd-valued column $j = 5$, the set of nat­ur­al num­bers will have the se­quence

all of which are evenly di­vis­i­ble by $5$. Wilson's the­o­rem is im­por­tant in analy­sis, but the rate at which the fac­to­r­i­al func­tion grows in size makes its use un­ten­able when cal­cu­lat­ing large prime num­bers and twin prime pairs. An ex­am­ple of the rapid rate at which the fac­to­r­i­al func­tion in­creas­es is seen for the prime number $p = 29$, for which $(29 - 1)!$has grown to ap­prox­i­mate­ly 305 oc­til­lion

De­spite pre­sent­ing the the­o­rem, nei­ther John Wilson nor his stu­dent Edward Waring could prove it. It was in 1771 that Joseph-Louis La­grange³ proved Wilson's The­o­rem, and ev­i­dence sug­gests that Got­tfried Wil­helm Leib­niz had proved the the­o­rem many years before 1771, which he did not pub­lish. Most modern-day treat­ments of Wilson's The­o­rem apply Fermat's Little The­o­rem to prove it.

The­o­rem (1.7.282) - Fermat's little the­o­rem: If $p$ is a prime number, for any in­te­ger $a$, the number $\left( a^{p} - a \right)$ is an in­te­ger mul­ti­ple of $p$, ex­pressed in mod­u­lar form as

By choos­ing $a = 4 \notin P$ and $p = 7 \in P$, we eval­u­ate the the­o­rem above

to find it equals the non-prime $a = 4.$The term co­prime de­fines that the only in­te­ger that di­vides both $a$ and $p$ is $1$. If $a$ is not di­vis­i­ble by $p$, it im­plies $a$ is co­prime to $p$, then Fermat's little the­o­rem is equiv­a­lent to the state­ment that $\left( a^{p - 1} - 1 \right)$ is an in­te­ger mul­ti­ple of $p$

which can be re­stat­ed as

We use the co­prime ap­pli­ca­tion of Fermat's little the­o­rem to the prime num­bers $p = 7$ and $a = 2$, which are, there­fore, co­prime

find­ing the re­sults holds. The ex­am­ple for $p = 8$ and $a = 2$, rep­re­sents two num­bers that are not co­prime and gives the result

which, unlike the pre­vi­ous, is non-zero. It is im­por­tant to re­mem­ber that two co­prime num­bers are not nec­es­sar­i­ly prime num­bers. For ex­am­ple, for $p = 4,\ a = 9$, we find the result is also $0$

The har­mon­ic num­bers $H_{n}$ are de­fined as the $n^{th}$ par­tial sum of the har­mon­ic series

From this series, the har­mon­ic number re­cur­rence re­la­tion­ship can be de­fined as

lead­ing to the har­mon­ic number dif­fer­ence iden­ti­ty

These iden­ti­ties can be used to derive di­vi­sion and prod­uct iden­ti­ties be­tween ad­ja­cent har­mon­ic num­bers

The har­mon­ic num­bers can be ap­plied an­a­lyt­i­cal­ly to Wilson's se­quence $a(n) = (n - 1)!(mod\ n)$, seen first by re­lat­ing $H_{n}$ to the fac­to­r­i­al $(n - 1)!$in the next straight­for­ward the­o­rem.

The­o­rem (1.7.295) $\mathbf{–}$ The fol­low­ing equiv­a­lence holds

Proof -

$\square$

There­fore, Wilson's se­quence $a(n) = (n - 1)!(mod\ n)$ can be ex­pressed in re­la­tion­ship to the har­mon­ic num­bers $H_{n}$ as

which holds ir­re­spec­tive of the con­gru­ent modulo op­er­a­tion. We can factor out $(n - 1)$ from the result above and divide both sides by $n$

then solve for $(n - 1)!$

Since

we have that

A re­cur­rence re­la­tion­ship can be seen by start­ing with the pre­vi­ous har­mon­ic number iden­ti­ty for $(n - 1)!$

which can be ex­pand­ed to the next fac­to­r­i­al

So, for $k \in \mathbb{Z}^{+}$, any fac­to­r­i­al $(n + k)!$is equiv­a­lent to the har­mon­ic number ex­pan­sion

The har­mon­ic num­bers are also re­lat­ed to the digam­ma func­tion $\psi^{(0)}(z)$, which is de­fined as the log­a­rith­mic de­riv­a­tive of the gamma func­tion $\Gamma(z)$

Since the gamma func­tion in­ter­po­lates the fac­to­r­i­al func­tion to com­plex and non-in­te­ger values, it im­plies for pos­i­tive in­te­ger values that

Like the fac­to­r­i­al func­tion, the gamma func­tion has the fol­low­ing re­cur­rence re­la­tion­ship

Specif­i­cal­ly, the har­mon­ic num­bers relate to the digam­ma func­tion $\psi^{(0)}(n)$ by the fol­low­ing

where $\gamma$ rep­re­sents the Euler-Mascheroni con­stant $\gamma$ The digam­ma func­tion holds to the fol­low­ing equiv­a­lence in­volv­ing the har­mon­ic num­bers and the Euler-Mascheroni con­stant

which leads to the abil­i­ty to define al­ter­nate forms of Wilson's the­o­rem using func­tions deeply rooted in analy­sis. We demon­strate this by taking Wilson's the­o­rem and ex­press­ing it in terms of the gamma func­tion and $p^{- 1}$

From the iden­ti­ty for $\psi^{(0)}(n + 1)$ from Eq. 1.7.308, we can ex­press $p^{- 1}$ as

to arrive at a form in­volv­ing the gamma func­tion, the har­mon­ic num­bers, the Euler-Mascheroni con­stant, and the digam­ma func­tion

A gen­er­at­ing set of a group $G$ de­fines a subset of el­e­ments $S \in G$, which under the group op­er­a­tion, typ­i­cal­ly mul­ti­pli­ca­tion or ad­di­tion, can ex­press every el­e­ment of $G$. For the subset $S \in G$, the sub­group of el­e­ments gen­er­at­ed by $S$ is de­fined as $\left\langle S \right\rangle$. In the case of $G = \left\langle S \right\rangle$, $S$ is said to gen­er­ate $G$, where the el­e­ments $\in S$ are de­fined as group gen­er­a­tors. If $S$ is equal to the empty set ∅, it im­plies that $\left\langle S \right\rangle$ is the iden­ti­ty $e$, oth­er­wise known as the triv­ial group $\left\{ e \right\}$.

If there is a lone el­e­ment $g \in S$, the sub­group $\left\langle S \right\rangle$ is gen­er­al­ly de­fined as $\left\langle g \right\rangle$, and im­plies that $\left\langle g \right\rangle$ is a cyclic sub­group gen­er­at­ed by the powers of $g$. By de­f­i­n­i­tion, cyclic groups are abelian, which sig­ni­fies the group op­er­a­tion is com­mu­ta­tive. The order of the cyclic group can then be es­tab­lished as $g = order$, which as­sumes a finite cyclic group. So, the order of finite $g$ is de­fined as the small­est pos­si­ble in­te­ger $n$ that obeys

and every order $n$ finite cyclic group is iso­mor­phic to the ad­di­tive group $\frac{Z}{nZ}.$If no in­te­ger $n$ es­tab­lish­es $g^{n} = 1$, the cyclic group is in­fi­nite in order, and iso­mor­phic to the ad­di­tive in­te­ger group $Z$, having $g = 1$ as its gen­er­a­tor by way of ad­di­tion. We com­pare and con­trast the finite and in­fi­nite order cyclic groups, where finite groups for the order $n = 5$ ex­am­ple, add by con­gru­ence modulo $5$

While the in­fi­nite order cyclic groups are the same up to $4$, but since mod­u­lar arith­metic is not em­ployed, they con­tin­ue as the set of in­fi­nite pos­i­tive in­te­gers

Being iso­mor­phic to the in­te­gers, an in­fi­nite order cyclic group has el­e­ments that are both pos­i­tive­ly and neg­a­tive­ly ori­en­tat­ed in re­la­tion­ship to the iden­ti­ty $\epsilon$, shown using the group al­ge­bra $G$ no­ta­tion

A finite cyclic group of order $n$ may be rep­re­sent­ed using the sim­i­lar al­ge­bra no­ta­tion above,

and an el­e­ment $g^{j}$ in the finite group will equal an­oth­er el­e­ment $g^{k}$, when the fol­low­ing holds for $g^{j} = g^{k}$

Finite cyclic groups may also be ex­pressed as roots of unity, where, for ex­am­ple the $5^{th}$ roots of unity are

which is iso­mor­phic to the group al­ge­bra of order $|G| = 5$

Given that the group's order is prime, $|G| =$ $p_{3} = 5$, the group is de­fined as simple, mean­ing it cannot be di­vid­ed into sub­groups. Prime-order finite cyclic groups are dis­tinct, given that any non-iden­ti­ty el­e­ment $\in G$ rep­re­sents a gen­er­a­tor of the group

The ro­ta­tion­al sym­me­tries of a poly­gon form a cyclic group, and if the set of sym­me­try ro­ta­tions of the said poly­gon equals $n$, it is iso­mor­phic to $\frac{Z}{nZ}$.

In group theory, the di­he­dral groups $D_{n}$ rep­re­sent the sym­me­tries of the reg­u­lar poly­gons, which in­clude the same ro­ta­tions in the plane found in the finite cyclic groups but also in­clude re­flec­tions. So, com­pared to the order $n$ from the cyclic groups, the di­he­dral group $D_{n}$ has $2n$ el­e­ments in its set, which de­fines it as having order $2n.$ $D_{n}$ can be rep­re­sent­ed as the set

where $r_{n}$ is the subset of ro­ta­tions and $s_{n}$ is the set of re­flec­tions across the lines of sym­me­try of the $n$-gon. So, in the $D_{5}$ set, $r_{n}$ is the $G_{n}$ finite cyclic group of order $n$.

The re­flec­tions $\left\{ s_{5} \right\}$ pre­sent­ed as oc­cur­ring by way of an or­thog­o­nal di­men­sion to the $r_{5}$-plane (right),

where the line of sym­me­try for the re­flec­tion runs from $g_{0} \rightarrow \frac{\left( g_{2} + g_{3} \right)}{2}$(yellow seg­ment).

In the figure above, we take the pen­ta­gon as a rigid body, having five ro­ta­tion­al sym­me­tries where ver­tices can rotate to any vertex ori­en­ta­tion in the plane. As a rigid body, a re­flec­tion re­quires that the pen­ta­gon come out of the plane to re­flect across an axis of sym­me­try of the pen­ta­gon. In the right image above, the dashed yellow cir­cles on the sphere's sur­face rep­re­sent the path the ver­tices take to ac­com­plish the re­flec­tion. The chosen axis of sym­me­try for the re­flec­tion above is rep­re­sent­ed by the seg­ment $\overline{g_{0}\left\lbrack \frac{\left( g_{2} + g_{3} \right)}{2} \right\rbrack}$, where $g_{0}$ re­mains fixed, while the pairs of ver­tices $\left\{ g^{2},g^{3} \right\},\left\{ g^{1},g^{4} \right\}$ swap po­si­tions.

Given that the har­mon­ic num­bers $H_{n}$ are in­ter­twined with the gamma and the fac­to­r­i­al func­tion, where $\Gamma(n + 1) = n!$, it be­comes more in­tu­itive when we rep­re­sent the nat­ur­al num­bers as reg­u­lar poly­gons (i.e., $n$-gons) cir­cum­scribed by the com­plex unit circle.

De­f­i­n­i­tion (1.7.322) - For any nat­ur­al number $n$, the Geo­met­ric Num­bers $\mathbf{G}_{\mathbf{n}}\mathbf{}$ are finite cyclic groups, rep­re­sent­ing the reg­u­lar $n$-gon cir­cum­scribed by the com­plex unit circle. The el­e­ments of $G_{n}$ are the set of ver­tices

where the first vertex is ori­ent­ed at the iden­ti­ty po­si­tion $e^{2i\pi\left( \frac{n}{n} \right)} = \epsilon$. $G_{n}$ rep­re­sents the $n^{th}$ roots of unity, and both the order of $G_{n}$ and the number of ver­tices of the $n$-gon cor­re­lates with the nat­ur­al number $n$. The ver­tices ori­ent­ed im­me­di­ate­ly coun­ter­clock­wise and clock­wise from the iden­ti­ty are always gen­er­a­tors of $G_{n}$,

De­f­i­n­i­tion (1.7.323) - For any prime number $p\in \mathbb{P}$, the Geo­met­ric Primes $\mathbf{G}_{\mathbf{p}}\mathbf{}$ are de­fined as the simple finite cyclic groups rep­re­sent­ing the reg­u­lar $p$-gon cir­cum­scribed by the com­plex unit circle, as the set of ver­tices

where the first vertex is ori­ent­ed at the iden­ti­ty po­si­tion $e^{2i\pi\left( \frac{p}{p} \right)} = \epsilon$. $G_{p}$ rep­re­sents the $p^{th}$ roots of unity, and both the order of $G_{p}$ and the number of ver­tices of the $p$-gon cor­re­lates with the prime number $p$. As a simple group, all el­e­ments $\in G_{p}$ are gen­er­a­tors of $G_{p}$, except the iden­ti­ty $\epsilon$

On the com­plex unit circle, the fol­low­ing equiv­a­lences hold

which im­plies that $G_{n}$ is equiv­a­lent to the set

Each Geo­met­ric Number $G_{n}$ has a cyclic group rep­re­sen­ta­tion $G_{n} = \frac{Z}{nZ}$, which can be rep­re­sent­ed by a square $n \times n$ matrix using the two gen­er­a­tors im­me­di­ate­ly clock­wise and coun­ter­clock­wise from the iden­ti­ty $\left\{ e^{\frac{(2i\pi}{n})},\ e^{\frac{( - 2i\pi}{n})} \right\}$.

In the cyclic matrix group form, the guar­an­teed pri­ma­ry gen­er­a­tors $\mathfrak{g}_{(1, + )},\mathfrak{g}_{(1, - )}$ are re­lat­ed by the matrix trans­pose

and these pri­ma­ry gen­er­a­tors will have eigen­val­ues equal to the $n$-gon vertex co­or­di­nates on the com­plex unit circle

Con­sid­er­ing the eigen­val­ue ma­tri­ces $D_{n}$ from the eigen­de­com­po­si­tion

these can be arranged in clock­wise and coun­ter­clock­wise pro­gres­sion along the di­ag­o­nal

where each eigen­val­ue on the di­ag­o­nal is a vertex of the $n$-gon. We start with the ex­am­ple of a non-prime nat­ur­al number $n = 4$ and the Geo­met­ric Number $G_{4}$ is com­posed of the fol­low­ing set of vertex el­e­ments

which rep­re­sents the ver­tices of the square/$4$-gon seen in Figure 1.7.31. The gen­er­a­tors in the $G_{4}$ set are $e^{2i\pi\left( \frac{1}{4} \right)}$ and $e^{2i\pi\left( \frac{3}{4} \right)}$, ev­i­denced by the fol­low­ing

In con­trast, the non-pri­ma­ry gen­er­a­tors $\in g_{4}$ are the el­e­ments $e^{2i\pi\left( \frac{2}{4} \right)}$ and $e^{2i\pi\left( \frac{4}{4} \right)}$,

where $e^{2i\pi\left( \frac{2}{4} \right)}$ rep­re­sent the prime number $g_{p(1)}$ in the prime de­com­po­si­tion of $n = 4$, and $e^{2i\pi\left( \frac{4}{4} \right)}$ is the iden­ti­ty el­e­ment, whose self-prod­ucts are itself $\left( e^{2i\pi\left( \frac{4}{4} \right)} \right)^{k} = e^{2i\pi\left( \frac{4}{4} \right)}.$

A finite cyclic group $G_{n}$ of order $n$ can be ex­pressed using the no­ta­tion $g_{n}^{k}$, where $g_{n}^{1}$ and $g_{n}^{- 1}$ are gen­er­a­tors, and $g_{n}^{k}$ is the $k^{th}$ el­e­ment in the group. Going left to right, we use the $G_{4}$ group ex­am­ple

in which $g_{4}^{1}$ and $g_{4}^{3}$ are the pri­ma­ry gen­er­a­tors

We can define a vector ${\widehat{G}}_{4}$ to en­cap­su­late the group el­e­ments as a whole

The ${\widehat{G}}_{4}$-vector can form a prod­uct with any el­e­ment in the group by left-mul­ti­pli­ca­tion, seen using the ex­am­ple $g_{4}^{1} \times {\widehat{G}}_{4}$

for which the pri­ma­ry gen­er­a­tor $g_{4}^{1}$ re­sults in vertex-spe­cif­ic ro­ta­tions that are pos­i­tive/coun­ter­clock­wise or­dered. This can be ex­pressed using ex­po­nen­tial func­tions as

which has an im­plic­it modulo-$4$ op­er­a­tion in the changes seen for the ra­tio­nal por­tions of the ex­po­nents. To ensure a clear un­der­stand­ing, we use the other pri­ma­ry gen­er­a­tor, $g_{4}^{3} = e^{2i\pi\left( \frac{3}{4} \right)} \in G_{n}$, to eval­u­ate its prod­uct with ${\widehat{G}}_{4}$

find­ing the ver­tices have ro­tat­ed in the neg­a­tive/clock­wise di­rec­tion, ev­i­denced by com­par­ing the ${\widehat{G}}_{4}$ vector to the re­sult­ing one, row-by-row. It is im­por­tant to be aware that a dif­fer­ence exists be­tween the per­ceived di­rec­tion of ro­ta­tion if one is con­sid­er­ing the square/$4$-gon as a rigid body ro­ta­tion, which would appear to rotate clock­wise for the $g_{4}^{1}$ prod­ucts, and coun­ter­clock­wise for the $g_{4}^{3}$ prod­ucts. Given its con­sis­ten­cy with the com­plex plane in­ter­pre­ta­tion of ro­ta­tions, the vertex-spe­cif­ic in­ter­pre­ta­tion is pre­ferred. Since the nat­ur­al number $n = 4$ has the prime de­com­po­si­tion $2^{2} = 4$, $g_{4}^{2}$ is the corol­lary el­e­ment to the prime $p_{1} = 2$.

which is seen to rotate each vertex by $i\pi$ ra­di­ans when ex­pressed in the ex­po­nen­tial func­tion form

Con­tin­u­ing with the $G_{4}$ ex­am­ple, the pos­i­tive pri­ma­ry gen­er­a­tors $g_{n}^{1} < g_{4}^{1}$ are the fol­low­ing

which, when taken to­geth­er as a prod­uct sum, reveal the Geo­met­ric Num­bers' re­la­tion­ship with the har­mon­ic series

The analog to eval­u­ate if $G_{4}$ is evenly di­vis­i­ble by values of $G_{n} < 4$ is by the fol­low­ing

which is pre­sent­ed in Table 1.7.3, show­ing that $g_{2}^{1}$ is a di­vi­sor of the group, which yields all the el­e­ments $\in G_{4}.$

$\displaystyle G_{4} \times g_{n}^{- 1}$	$\displaystyle g_{4}^{0}$	$\displaystyle g_{4}^{1}$	$\displaystyle g_{4}^{2}$	$\displaystyle g_{4}^{3}$
$\displaystyle g_{1}^{- 1}$	$\displaystyle g_{(4,2)} = e^{2i\pi\left( \frac{0}{4} \right)}$	$\displaystyle g_{(4,1)} = e^{2i\pi(1/4\ )}$	$\displaystyle g_{(4,2)} = e^{2i\pi\left( \frac{2}{4} \right)}$	$\displaystyle g_{(4,3)} = e^{2i\pi\left( \frac{3}{4} \right)}$
$\displaystyle g_{2}^{- 1}$	$\displaystyle g_{(4,2)} = e^{2i\pi\left( \frac{2}{4} \right)}$	$\displaystyle g_{(4,3)} = e^{2i\pi(3/4\ )}$	$\displaystyle g_{(4,0)} = e^{2i\pi\left( \frac{0}{4} \right)}$	$\displaystyle g_{(4,1)} = e^{2i\pi\left( \frac{1}{4} \right)}$
$\displaystyle g_{3}^{- 1}$	$\displaystyle g_{(12,8)} = e^{2i\pi\left( \frac{8}{12}\ \right)}$	$\displaystyle g_{(12,11)} = e^{2i\pi(11/12\ )}$	$\displaystyle g_{(12,2)} = e^{2i\pi\left( \frac{2}{12}\ \right)}$	$\displaystyle g_{(12,5)} = e^{2i\pi\left( \frac{5}{12}\ \right)}$

At­ten­tion should be di­rect­ed to the fact that prod­uct $g_{4} \times g_{(n,1)}^{- 1}$ yields an or­dered se­quence of el­e­ments $\in g_{4}$ when going from left to right in the row of the table. This or­dered se­quence is equiv­a­lent to the ex­po­nen­tial func­tion prod­uct with our ${\widehat{G}}_{4}$ vector

and rep­re­sents the even di­vis­i­bil­i­ty of $p_{1} = 2$ for the nat­ur­al number $n = 4$. In­tu­itive­ly, this is the case given that

which is a number ro­tat­ing in the op­pos­i­tive di­rec­tion but is equal to the $g_{(4,2)}$ el­e­ment

The ex­am­ple of the prime nat­ur­al number $n = 7 \in \mathbb{P}$ rep­re­sents a Geo­met­ric Number in the Geo­met­ric Primes $G_{7} \in G_{\mathbb{P}}$. The $G_{7}$ group is com­posed of the fol­low­ing set of vertex el­e­ments

which rep­re­sents the ver­tices of the reg­u­lar $7$-gon seen in Figure 1.7.32. Since $G_{7} \in G_{\mathbb{P}}$, it is a simple group, mean­ing all of its gen­er­a­tors $g_{7}^{k}$ are pri­ma­ry, except for the iden­ti­ty el­e­ment $g_{7}^{0}$

Any $g_{7}^{k}$ gen­er­a­tor can form a prod­uct with the $G_{7}$ group vector ${\widehat{G}}_{7}$

which acts to rotate each vertex of the $7$-gon by the in­te­ger value for $k$.

Wilson's the­o­rem, we recall, states that all prime num­bers $p$ obey the fol­low­ing

We can con­sid­er how the geo­met­ric primes relate to the the­o­rem by set­ting a basic foun­da­tion per­tain­ing to the even­ness and odd­ness of prod­ucts and quo­tients of the nat­ur­al num­bers.

Lemma (1.7.356) - Nat­ur­al num­bers obey the fol­low­ing even/odd rules as prod­ucts

For evenly di­vis­i­ble quo­tients, $a,b\in \mathbb{N}$, where $a > b$, the fol­low­ing holds for even $a_{e},b_{e}$,

and for even $a$, and odd $b$ $\left( a_{e},b_{o} \right)$

For evenly di­vis­i­ble odd $a,b$, where $a > b\in \mathbb{N}$, we have

and there would exist no evenly di­vis­i­ble quo­tients formed by odd $a$ with even $b$.

Lemma (1.7.357) - The prod­uct of $k$ var­i­ous nat­ur­al num­bers $N_{k}$ will be even if and only if there exist at least one even-value $n_{j(e)}$ in the prod­uct

This is true, given the dis­trib­u­tiv­i­ty of the prod­uct op­er­a­tion, we can group the odd and even $n_{j}$ into the fol­low­ing prod­uct

For $n_{j(e)} \neq \varnothing$, using the even/odd prod­uct no­ta­tion, $N_{k} \in \mathbb{N}_{e}$

and for $n_{j(e)} = \varnothing$, we find $N_{k} \in \mathbb{N}_{o}$

$\square$

The fac­to­r­i­al func­tion $n! \in \mathbb{N}$ is equal to the prod­uct of the con­sec­u­tive nat­ur­al num­bers from $1$ to $n$

which by the dis­trib­u­tiv­i­ty of the prod­uct op­er­a­tion also sep­a­rates into the prod­uct of odd and even sets of nat­ur­al num­bers

Lemma (1.7.360) – For a nat­ur­al number $n\notin \mathbb{P}$, the fac­to­r­i­al func­tion $(n - 1)!$ is evenly di­vis­i­ble by $n$ and even $\forall\ n > 4$

The com­pos­ite number $n$ is the prod­uct of at least two primes raised to var­i­ous powers$\ \in (n - 1)!$, and there­fore $\frac{(n - 1)!}{n}$ is an evenly di­vis­i­ble quo­tient. The first even com­pos­ite number $> 4$ is $6$, which for $N_{6,e}$ is the fol­low­ing

and $n$ has the prime number com­po­si­tion $n = 3 \bullet 2$. The $p_{1},p_{2}$ primes cancel in the ra­tio­nal, but the even-valued $p_{1}^{2} = 4$ re­mains. There­fore, the even set of re­main­ing $N_{(k,e)} \neq \varnothing$, which as­sures the result is even

The same holds for odd com­pos­ite nat­ur­al num­bers, the first of which is $n = 9$, having the quo­tient form

where $n = 3 \bullet 3$ evenly di­vides $(n - 1)!,$ and ac­tu­al­ly adds an­oth­er even nat­ur­al number to $N_{9,e}$

For in­creas­ing $n$, more even nat­ur­al num­bers pop­u­late the fac­to­r­i­al $(n - 1)!$ , and the even nat­ur­al number set $\in N$ is always non-empty, $N_{e} \neq \varnothing$.

The­o­rem (1.7.361) – For all prime num­bers $p_{k}\in \mathbb{P}$, the func­tion $N_{p}$ is not evenly di­vis­i­ble, taking the ra­tio­nal form

Proof – Given that $p$ is prime and greater in value than any other prime in the de­com­po­si­tion of $(p - 1)!$, it im­plies there are no even di­vi­sors of $p \in (p - 1)!$. There­fore, $N_{p}$ is not evenly di­vis­i­ble, and given that $n > (n - 1)$, the func­tion takes the re­sul­tant form

$\square$

By the pre­vi­ous lemmas and the­o­rem, we can now prove the anal­o­gy to Wilson's the­o­rem for the Geo­met­ric Primes $g_{p}$.

The­o­rem (1.7.362) – The Geo­met­ric Primes $g_{P}$ obey the fol­low­ing ex­pres­sion for all prime num­bers $p\in \mathbb{P}$

and for all $n\notin \mathbb{P} > 4$, they obey the equiv­a­lence

Proof. – The rep­re­sen­ta­tion of Euler's for­mu­la

con­tains the cosine and sine func­tions, which are by de­fault con­gru­ent modulo $2\pi$. Set­ting $\theta = \left( \frac{2\pi}{n}(n - 1)! \right)$, the ex­po­nen­tial func­tion $e^{i\theta}$ be­comes

The ex­po­nen­tial vari­able can be rewrit­ten as

where the fol­low­ing holds $\forall\ n$

and by ap­ply­ing the modulo $2\pi$ con­gru­en­cy, the $e^{i\left( \frac{2\pi}{n}(n - 1)! \right)}$ func­tion can be writ­ten as

With the ex­cep­tions of $0!,1!$, by the pre­vi­ous lemmas, the fac­to­r­i­al func­tion $n!$ will always be even-valued. For $n > 4$, the com­pos­ite nat­ur­al num­bers, $n\neq \mathbb{P},$ will have a prime number com­po­si­tion $\in (n - 2)!$, which can­cels the $\frac{1}{n}$ term, im­ply­ing the fol­low­ing holds for $n > 4$

where $n$ is the com­po­si­tion of primes $\in (n - 1)!$, and by the pre­vi­ous lemma, the term in­volv­ing the fac­to­r­i­al quo­tient $N_{e}$ will always remain even. There­fore, the ex­po­nent con­gru­ent modulo $2\pi$ always equals $0$

given that $2\pi$ evenly di­vides $2\pi N_{e}$. By the pre­vi­ous the­o­rem $\forall\ p\in \mathbb{P},$ the func­tion

is not evenly di­vis­i­ble. By the lemma, $(p - 2)!$ is even $\forall\ p > 4$, and there­fore, $(p - 2)!$ is evenly di­vis­i­ble by $2$ $\forall\ p > 4$, having a quo­tient $q\in \mathbb{N}$ with­out a re­main­der. Given that

this rep­re­sents the ir­re­ducible prod­uct term in the con­gru­ent modulo $2\pi$ op­er­a­tion, which taken to­geth­er with $(p - 2)!$, im­plies that

Plug­ging these vari­ables back into the ex­po­nen­tial func­tion, it holds for $p > 4 \in \mathbb{P}$ that

which rep­re­sents the first pri­ma­ry gen­er­a­tor $\in g_{p}$ sit­ting clock­wise from the iden­ti­ty el­e­ment on the com­plex unit circle.

For $n > 4 \notin \mathbb{P}$, we have proved that $\left\lbrack 2\pi N_{e}\ \ mod\ 2\pi \right\rbrack = 0$, im­ply­ing the ex­po­nen­tial form $\forall\ n\notin \mathbb{P} > 4$ equals

$\square$

De­f­i­n­i­tion (1.7.363) – The nat­ur­al non-prime Geo­met­ric Num­bers $g_{n}$ $(n\neq \mathbb{P})$ obey the fol­low­ing ex­pres­sion

except for $n = \left\{ 4 \right\}$, where the fol­low­ing holds

In the com­plex rep­re­sen­ta­tion, the lo­ca­tions of the geo­met­ric primes can be ex­pressed as

where each prime co­or­di­nate $p\in \mathbb{N},\mathbb{P}$ co­or­di­nate is plot­ted in para­met­ric form over the reals in Figure 1.7.34, using the para­met­ric ex­pres­sion

In Figure 1.7.34, the $g_{t}$ wave­form is plot­ted in con­junc­tion with the co­in­cid­ing para­met­ric fac­to­r­i­al ex­pres­sion over the reals

where all the prime num­bers will be lo­cat­ed on the $\cos\left( \frac{2\pi}{p} \right)$ curve at the points of in­ter­sec­tion with the $\cos\left( \frac{2\pi(n - 1)!}{n} \right)$ wave­form.

The con­gru­ent modulo ex­pres­sion from Wilson's the­o­rem can be ap­plied to the cyclic matrix rep­re­sen­ta­tion of our $g_{(4,1)}$ pri­ma­ry gen­er­a­tor

and given that

we see the matrix form is con­sis­tent with the ex­pres­sion for $n = 4$

Figure 1.7.35 rep­re­sents the para­met­ric func­tions for $g_{p}$ (red) and $e^{\left( \frac{2i\pi}{n}\left\lbrack (n - 1)! \right\rbrack \right)}$ (blue) plot­ted in the com­plex plane rising along $z = n$

The first five prime num­bers rep­re­sent the in­ter­sec­tion of the two para­met­ric curves (blue/red co­or­di­nates), and the non-prime nat­ur­al num­bers do not rep­re­sent in­ter­sec­tions on the two curves (light/dark green co­or­di­nates).

1. Edward Waring, "Med­i­ta­tiones Al­ge­braicae" (Cam­bridge, Eng­land: 1770), p. 218
2. https://mathshis­to­ry.st-an­drews.ac.uk/Bi­ogra­phies/Al-Haytham/
3. Joseph Louis La­grange, "Demon­stra­tion d'un théorème nou­veau con­cer­nant les nom­bres pre­miers", Nou­veaux Mé­moires de l'Académie Royale des Sci­ences et Belles-Let­tres (Berlin), vol. 2, pages 125–137 (1771).