# Primal Conjecture in Matrix $E_{\phi}^{9}$.

Paul Marrero
<EMAIL_ADDRESS>
&Eduardo J. Acuña T.
<EMAIL_ADDRESS>

(April 2021)

###### Abstract

_"There is a certain enigma in everything that seems obvious."_

In this paper we propose a conjecture about integer solutions to any
equations, based on Primal algebra specifically this conjecture is a corollary
of the Acuña´s Theorem in that article.
Also some problems are proposed which, if the conjecture is correct, could be
solved.

##  1 Introduction

The approach of this article is from the mathematical philosophical position
of constructivism, that is to say that we create an arithmetic system with our
rules and we observe how the elements behave and what relation they have with
the real mathematics.
With this in mind we introduce Primordial Algebra as a system of analysis
aimed at integers, especially operations between integers $\pm\mathbb{Z}$.
This algebra analyzes the numbers under a unitary module which is the 9 in
base 10, in this way the remainders of each positive or negative integer
generate groups congruent to each other, although these properties have been
studied in the past it had not been possible to operate between positive and
negative, the primordial algebra allows to operate between both types without
falling into incongruities.

$\displaystyle\frac{-Z}{-9Z}\leavevmode\nobreak\ \bigcup\leavevmode\nobreak\
\frac{Z}{9Z}\leavevmode\nobreak\ /\leavevmode\nobreak\
Z\in\leavevmode\nobreak\ \mathbb{Z}^{+},\leavevmode\nobreak\
-Z\leavevmode\nobreak\ \in\leavevmode\nobreak\ \mathbb{Z}^{-}$

###  1.1 Notation

Before beginning to understand how primordial algebra works, it is necessary
to understand the notation created for it.

  * •

Let $v$ be any integer number

  * •

Let $\Phi$ be a function of the digital root of any $\mathbb{Z}$

  * •

Let $\Phi^{Z}$ be any of the class of any $\mathbb{Z}$, we replace $"Z"$ to
denote the value of the class.

  * •

Let $Z_{\phi}$ be any Primal Number, which is the representative value of the
class and can be operated with.

###  1.2 Reduction of integers

The First idea that we use for this matrix was the property of reduction of
integer numbers by the Digital Sum, all this is mention in the article (2021)
[5], This is congruent whit the Method of Digital Sum Propose by Jhon Eider
(2015) [3], and we propose the the next Equation to found any Positive Digital
Root based on Knuth (1972) [2].
Also this paper is gonna drop a resume of primal algebra, with the propose of
make more easy to understand the conjecture.

The Novem Digital Root:

$\Phi_{9}=v-9\left(\left\lfloor\frac{v-1}{9}\right\rfloor\right)$ (1)

in the Matrix $E_{\phi}^{9}$ the equation for the Digital Sum is:

$\Phi_{j}=v-9(i-1)$ (2)

Both of this equations are equivalent, so we conclude when the digital root is
in module 9, is equal to the Digital Sum.

##  2 Theorem Marrero Criterion

The Marrero’s criterion, (2021) [5], also mentioned in the pre-print [1] as
the $K$-interactive of the digital Sum. This is a criterion who tell when to
stop to the Digital Sum $\Phi$, we denoted the Criterion like "$M$" this is
the mark to any function of Digital Sum that meet the condition of the
criterion $\phi^{M}$.

###  2.1 Theorem M

Let $v$ are integer, then there is a unique integer $d\in\hskip 2.84544ptD$
such that:

$\phi^{k}(v_{k})-d=0,$ (3)

where $D$ of the set of the points fixed of $v$.

###  2.2 Proof:

  1. 1.

(Existence). Supposed $b\in\mathbb{Z}$ and $k$ be a integer fixed. Consider
the set $S\left\\{b=\phi^{k}(v_{k})-b=0\right\\}$ where $v_{k}\in\mathbb{Z}$,
$S$ is not empty, because $1\in S$ with $k=1$. Then $S\subseteq\mathbb{Z}$ and
let $d$ being the smallest element of $S,d\in S$, it must have the form
$\phi^{k}(v_{k})-d=0$, therefore $\phi^{k}(v_{k})=d.$

  2. 2.

(Uniqueness). Let $d_{1}$ and $d_{2}\in D$ such that $d_{1}\neq d_{2}$.
supposed that $\phi^{k_{1}}(v_{k})-d_{1}=0$ and $\phi^{k_{2}}(v_{k})-d_{2}=0$
(using Definition 2.1), then
$\phi^{k_{1}}(v_{k})-d_{1}=\phi^{k_{2}}(v_{k})-d_{2},$ but
$\phi^{k_{1}}(v_{k})-\phi^{k_{2}}(v_{k})=d_{1}-d_{2}$ and
$\phi^{k_{1}}(v_{k})-\phi^{k_{2}}(v_{k})=0$. therefore $d_{1}=d_{2}$, this
constitutes a contradiction. $\square$

##  3 Elements of the Matrix

In this section we give a brief introduction to the matrix $E_{\phi}^{9}$, it
is formed by the union of two positive and negative sub-matrices together with
its own classification system, we adopt the following convention: Primal
number $Z_{\phi}$ ,Primal class $\Phi^{Z}$, integer numbers $v$, rows $(i)$,
columns $(j)$. See the following figures for more details.

$\begin{bmatrix}1_{\phi}&2_{\phi}&3_{\phi}&4_{\phi}&5_{\phi}&6_{\phi}&7_{\phi}&8_{\phi}&9_{\phi}\\\
1&2&3&4&5&6&7&8&9\\\ 10&11&12&13&14&15&16&17&18\\\
19&20&21&22&23&24&25&26&27\\\ 28&29&30&31&32&33&34&35&36\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\end{bmatrix}$

Table 1: _Positive sub-matrix_ $[V^{+}]$

$\begin{bmatrix}-1_{\phi}&-2_{\phi}&-3_{\phi}&-4_{\phi}&-5_{\phi}&-6_{\phi}&-7_{\phi}&-8_{\phi}&-9_{\phi}\\\
-1&-2&-3&-4&-5&-6&-7&-8&-9\\\ -10&-11&-12&-13&-14&-15&-16&-17&-18\\\
-19&-20&-21&-22&-23&-24&-25&-26&-27\\\ -28&-29&-30&-31&-32&-33&-34&-35&-36\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\end{bmatrix}$

Table 2: _Negative sub-matrix_ $[V^{-}]$

###  3.1 Primal Numbers

The primal number is the result of applying the digital sum function $\Phi$
and we will denote it by $Z_{\phi}$. We denote the set of positive and
negative primal numbers by

$\begin{matrix}D=\left\\{{{d|-9_{\phi}\leq d\leq
9_{\phi}|}}\right\\}\end{matrix}$

Note that each value of $D$ represents a relationship between and its classes,
that is, for every integer $v$ that belongs to column $j$, there exists a
primal number $Z_{\phi}$ such that $Z_{\phi}$ is congruent with $j$ module 9.

####  3.1.1 Example

  1. 1.

$9_{\phi}=72$

  2. 2.

$8_{\phi}=179$

  3. 3.

$4_{\phi}=9211$
⋮

  4. 4.

$-5_{\phi}=-14$

  5. 5.

$-9_{\phi}=-927$

###  3.2 The Null Primal class

The Null class $\emptyset$, we will denote it by 0, is the one that is
constituted by a column of zeros, and establishes a numerical border between
the positive and negative classes of the matrix $E_{\phi}^{9}$.

$\begin{bmatrix}-9_{\phi}&-8_{\phi}&\ldots&-1_{\phi}&\emptyset&1_{\phi}&2_{\phi}&\ldots&9_{\phi}\\\
-9&-8&\ldots&-1&0&1&2&\ldots&9\\\ -18&-17&\ldots&-10&0&10&11&\ldots&18\\\
-27&-26&\ldots&-19&0&19&20&\ldots&27\\\
-36&-35&\ldots&-28&0&28&29&\ldots&36\\\
-45&-44&\ldots&-37&0&37&38&\ldots&45\\\
-54&-53&\ldots&-46&0&46&47&\ldots&54\\\
-63&-62&\ldots&-55&0&55&56&\ldots&63\\\
-72&-71&\ldots&-64&0&64&65&\ldots&72\\\
-81&-80&\ldots&-73&0&73&72&\ldots&81\\\
-90&-89&\ldots&-82&0&82&83&\ldots&90\\\
-99&-98&\ldots&-91&0&91&92&\ldots&99\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
\end{bmatrix}$

Table 3: Matrix $E_{\phi}^{9}$

###  3.3 Primal Class

The primal class that we denote by $\Phi^{Z}$ represents and are sets of
congruence classes modulo 9, that is:

$\Phi^{Z}=\left\\{v\in\mathbb{Z}\mid Z_{\phi}\equiv v\hskip
2.84544pt(mod\leavevmode\nobreak\ 9)\right\\}$

####  3.3.1 Example:

  1. 1.

$\Phi^{1}\hskip 5.69046pt=\left\\{1,10,19,28,...,91...\right\\}$

  2. 2.

$\Phi^{2}\hskip 5.69046pt=\left\\{2,11,20,29,...,92...\right\\}$

  3. 3.

$\Phi^{9}\hskip 5.69046pt=\left\\{9,18,27,36,...,99...\right\\}$

  4. 4.

$\Phi^{-1}=\left\\{-1,-10,...,-199...\right\\}$

  5. 5.

$\Phi^{-9}=\left\\{-9,-18,...,-999...\right\\}$

The Primal class also are class of congruence with respect to the module 9,
Vinogradov (1977) [6], in class $9^{+}_{-}$ the residue is 0, but we take the
module as the main value of the class, either positive or negative.

##  4 Method of calculation of the positive columns

Recall that the columns of the matrix $E_{\phi}^{9}$, we denote it by $j$, it
is worth mentioning that said Matrix $E_{\phi}^{9}$,consists of 18 columns,
nine are for the positive part $[V^{+}]$ and nine for the negative part
$[V^{-}]$ along with its respective primal number that we write at the
beginning of each column in the matrix $E_{\phi}^{9}$.

There is many methods to calculate the columns by definition we chose the
Digital Sum $\Phi$, The novem digital$\Phi_{9}$1 or The matrix form
$\Phi_{j}$2. All mentioned before are valid to the positive Part $[V^{+}]$.

##  5  Method of calculation of the Negative Columns

As the same way we define the positive columns, the negative part of the
matrix contains all the negative integers $\mathbb{Z}^{-}$, this calculate is
possible thanks to the negative novem digital root.

###  5.1 Definition:

Let $v$ be a negative integer, we will write it as $-v$. We define the
negative digital root novem, which we denote by $\mho$ and is as follows:

$\mho_{-9}(v)=((v+1)\leavevmode\nobreak\ mod\hskip 2.84544pt(-9))-1$ (4)

###  5.2 Definition 2:

We define the negative novem floor function,[4] for a negative integer, we
will denote it by and it is as follows:

$\hat{\mho}_{-9}(v)=v-(-9){\left\lfloor{\frac{v+1}{-9}}\right\rfloor}$ (5)

For the Negative Digital Sum in the matrix $\Phi_{-j}$ we propose the next
equation:

$\Phi_{-j}=v+9(i-1)$ (6)

Where equation 5 and 6 are congruent with each other.

##  6 Method of Calculation of the rows of the Matrix.

Let $i$ be the set of rows of the Matrix $E_{\phi}^{9}$, the rows of the
Matrix are always positive for any number inside the matrix, no matters if is
negative, but the method to calculate their $i$ is conditional to their sing
of the number.

To calculate the rows of the Matrix $E_{\phi}^{9}$ we proceed to consider two
cases:

  * •

case 1: Row a positive integer.

    1. $i_{v}=\left\\{\begin{array}[]{c}\left\lfloor\frac{v}{9}+1\right\rfloor\quad\leavevmode\nobreak\ For\leavevmode\nobreak\ a\leavevmode\nobreak\ v\neq 9c\\\ \vspace{0.2cm}\hfil\\\ \frac{v}{9}\hskip 34.14322pt\leavevmode\nobreak\ For\leavevmode\nobreak\ a\leavevmode\nobreak\ v=9c\end{array}\right.$ (7)

  * •

case 2: Row a negative integer.

    1. $i_{-v}=\left\\{\begin{array}[]{c}\left\lfloor-\frac{v}{9}+1\right\rfloor\quad\leavevmode\nobreak\ For\leavevmode\nobreak\ a\leavevmode\nobreak\ v\neq 9c\\\ \vspace{0.2cm}\hfil\\\ -\frac{v}{9}\hskip 34.14322pt\leavevmode\nobreak\ For\leavevmode\nobreak\ a\leavevmode\nobreak\ v=9c\end{array}\right.$ (8)

Where $9c$ are all the multiple of 9.

Other ways to calculate the $i$ is whit this two equations:

$i_{v}=\frac{v-\phi_{v}}{9}+1$ (9)

$i_{-v}=\frac{v-\phi_{-v}}{9}-1$ (10)

##  7 Decompress Primal numbers

One of the advantage of putting the integers numbers in the Matrix
$E_{\phi}^{9}$ and turning into Primal numbers is that every number no matter
the size can be compress into one number, but whit the matrix equations we can
decompress those Primal numbers and get back the integer number. This is the
fundamental base for the creation of a algebra of from the matrix, because the
equations to decompress are getting isolating the formulas 2 and 6:

Isolating equation 2 to get the positive number we get:

$v=\Phi_{j}+9(i-1)$ (11)

Isolating equation 6 to get the negative number we get:

$-v=\Phi_{-j}-9(i-1)$ (12)

###  7.1 Notation for reduced numbers in the matrix.

As we know a integer number in the Matrix is a set of three elements, his
Primal Number $Z_{\phi}$, his position in $j$ and $i$, whit this we can
represent a integer inside the matrix like this:

$\displaystyle v=Z_{i}$

Where $i$ is the row and $Z$ is the representative number, we don’t use $\phi$
because would be redundant such that $\phi\equiv Z\equiv j$ are numerically
the same, so we prefer pick only one to represent all.

####  7.1.1 Example:

Now some examples of how represent numbers in the matrix.

  1. 1.

$v=178,\rightarrow Z_{i}=7_{20}$

  2. 2.

$v=157,\rightarrow Z_{i}=4_{18}$

  3. 3.

$v=293,\rightarrow Z_{i}=5_{33}$

  4. 4.

$v=359,\rightarrow Z_{i}=8_{40}$

  5. 5.

$v=999,\rightarrow Z_{i}=9_{111}$

If we go to the Matrix and locate the $Z$ in the columns and the $i$ in the
rows the interception would be the correspondent number. Now whit the
decompress equation we can go from $Z_{i}$ to $v$ using the example 4. shown
previously.

$\displaystyle\begin{matrix}8_{40}=8+9(40-1)\rightarrow 8_{40}=8+9(39)\\\
8_{40}=8+351\rightarrow 8_{40}=359\\\ \end{matrix}$

##  8 The Acuña’s Theorem

The Acuña’s Theorem [5], is the core theorem that expose the principles of the
algebra primordial, using the properties of the matrix to arithmetic between
classes.

###  8.1 Definition:

For all operation that we denote as $(\odot=+,-,/,*)$ between two or more
Primal Classes, the result will be another Primal class, and this is congruent
with the numbers inside the class.

$\displaystyle\forall\hskip 5.69046ptZ_{\phi a}\odot Z_{\phi b}\equiv
v_{a}\odot\hskip 2.84544ptv_{b}\leavevmode\nobreak\ (mod\leavevmode\nobreak\
9)$

This is a extension of of class of residues by Vinogradov (1977) [6], but this
work in both positive and negative integers.

###  8.2 Proof:

If:

$\displaystyle\begin{matrix}\vspace{0.2cm}v_{a},Z_{\phi
a}\in\phi^{a}\rightarrow v_{a}\equiv Z_{\phi a}\leavevmode\nobreak\ (mod\hskip
2.84544pt9)\\\ \vspace{0.2cm}v_{b},Z_{\phi b}\in\phi^{b}\rightarrow
v_{b}\equiv Z_{\phi b}\leavevmode\nobreak\ (mod\hskip 2.84544pt\hskip
2.84544pt9)\\\ v_{c},Z_{\phi c}\in\phi^{c}\rightarrow v_{c}\equiv Z_{\phi
c}\leavevmode\nobreak\ (mod\hskip 2.84544pt9)\end{matrix}$

Then by definition:

$\displaystyle\begin{matrix}(\phi\odot\phi)^{a+b=}=\left\\{x\in\mathbb{Z}\mid
x\equiv Z_{\phi a}+Z_{\phi b}=Z_{\phi c}\leavevmode\nobreak\ (mod\hskip
2.84544pt9)\right\\}\end{matrix}$

From where:

$\displaystyle\begin{matrix}\vspace{0.2cm}x-(Z_{\phi a}+Z_{\phi b}=Z_{\phi
c})=9t,\hskip 5.69046ptfor\hskip 2.84544ptt\in\mathbb{Z}\\\
\vspace{0.2cm}x=9t+(Z_{\phi a}+Z_{\phi b}=Z_{\phi c})\\\
\vspace{0.2cm}x=9t+[(v_{a}-9q)+(v_{b}-9r)=(v_{c}-9s)]\hskip 5.69046ptfor\hskip
5.69046ptq,r,s\in\mathbb{Z}\\\
\vspace{0.2cm}x=9t+[(v_{a}+v_{b})-9(q+r)=v_{c}-9s]\\\
\vspace{0.2cm}x=9t+[(v_{a}+v_{b})-9(q+r)+9s=v_{c}]\\\
\vspace{0.2cm}x=9t+[9(-q-r+s)+(v_{a}+v_{b})=v_{c}]\\\
\vspace{0.2cm}x=9(-q-r+s+t)+(v_{a}+v_{b})=v_{c}\\\ \rightarrow x\equiv
v_{a}+v_{b}=v_{c}\hskip 5.69046pt(mod\hskip 2.84544pt9)\therefore
x\in(\phi\odot\phi)^{a+b=c}.\hskip 5.69046pt\square\end{matrix}$

##  9 Group Macro Primal ($\zeta_{\phi},\circledcirc$)

We define the group Macro Primal is not empty and have the next properties:

###  9.1 Proof:

Let Macro Primal be:

$\displaystyle\left\\{\zeta_{\phi}\right\\}=\left\\{\Phi^{Z}\mid Z\in
D\right\\}\wedge\left\\{0\right\\}$

####  9.1.1 Is closed for operation:

$\displaystyle\phi^{a}\circledcirc\phi^{b}=(\phi\circledcirc\phi)^{(a\circledcirc
b)}$

Let’s suppose

$\displaystyle(\phi\circledcirc\phi)^{(a\circledcirc
b)}=(\phi^{\prime}\circledcirc\phi^{\prime})^{(a^{\prime}\circledcirc
b^{\prime})}$

Then if

$\displaystyle\begin{matrix}\vspace{0.2cm}Z_{\phi a},Z^{\prime}_{\phi
a}\in\phi^{a}\rightarrow Z_{\phi a}\equiv Z^{\prime}_{\phi
a}\leavevmode\nobreak\ (mod\hskip 2.84544pt9)\\\ \vspace{0.2cm}Z_{\phi
b},Z^{\prime}_{\phi b}\in\phi^{b}\rightarrow Z_{\phi b}\equiv Z^{\prime}_{\phi
b}\leavevmode\nobreak\ (mod\hskip 2.84544pt9)\\\ \vspace{0.2cm}\therefore\\\
\vspace{0.2cm}9\mid Z_{\phi a}-Z^{\prime}_{\phi a}\rightarrow Z_{\phi
a}-Z^{\prime}_{\phi a}=9q/q\in\mathbb{Z}\\\ \vspace{0.2cm}9\mid Z_{\phi
b}-Z^{\prime}_{\phi b}-Z_{\phi a}-Z^{\prime}_{\phi b}\rightarrow Z_{\phi
b}-Z^{\prime}_{\phi b}=9s/s\in\mathbb{Z}\end{matrix}$

By definition we have to:

$\displaystyle(\phi\circledcirc\phi)^{(a\circledcirc
b)}=\left\\{v\in\mathbb{Z}\mid(Z_{\phi a}\circledcirc Z_{\phi
b})\leavevmode\nobreak\ (mod\hskip 2.84544pt9)\right\\}$

Then we verified that

$\displaystyle\begin{matrix}\vspace{0.2cm}9\mid v-(Z_{\phi a}\circledcirc
Z_{\phi b})\\\ v-(Z_{\phi a}\circledcirc Z_{\phi
b})=9t/t\in\mathbb{Z}\end{matrix}$

then

$\displaystyle\lambda)v=9t+(Z_{\phi a}\circledcirc Z_{\phi b})$

Starting from $\lambda$ we will continue the proof for the addition operation
and later we will continue from it for multiplication.

$\displaystyle\begin{matrix}\vspace{0.2cm}v=9t+(Z_{\phi a}+Z_{\phi b})\\\
\vspace{0.2cm}v=9t+(9q+Z^{\prime}_{\phi a}+9s+Z^{\prime}_{\phi b})\\\
v=9(t+q+s)+(Z^{\prime}_{\phi a}+Z^{\prime}_{\phi b})\end{matrix}$

How $(t+q+s)\in\mathbb{Z}$, we finally get that:

$\displaystyle\begin{matrix}\vspace{0.2cm}v=9(t+q+s)+(Z^{\prime}_{\phi
a}+Z^{\prime}_{\phi b})\rightarrow v\equiv(Z^{\prime}_{\phi
a}+Z^{\prime}_{\phi b})\leavevmode\nobreak\ (mod\leavevmode\nobreak\ 9)\\\
\vspace{0.2cm}\therefore\\\ v\in(Z^{\prime}_{\phi a}+Z^{\prime}_{\phi
b})\end{matrix}$

Now we continue the proof from $\lambda$ for the multiplication:

$\displaystyle\begin{matrix}\vspace{0.2cm}v=9t+(Z^{\prime}_{\phi
a}*Z^{\prime}_{\phi b})\\\ \vspace{0.2cm}v=9t+[(9q+Z^{\prime}_{\phi
a})(9s+Z^{\prime}_{\phi b})]\\\ \vspace{0.2cm}v=9t+[(81qs+9qZ^{\prime}_{\phi
b}+9sZ^{\prime}_{\phi a}+Z^{\prime}_{\phi a}*Z^{\prime}_{\phi b})]\\\
\vspace{0.2cm}v=(9t+81qs+9qZ^{\prime}_{\phi b}+9sZ^{\prime}_{\phi
a})+(Z^{\prime}_{\phi a}*Z^{\prime}_{\phi b})\\\
\vspace{0.2cm}v=9(t+9qs+9qZ^{\prime}_{\phi b}+9sZ^{\prime}_{\phi
a}+(Z^{\prime}_{\phi a}*Z^{\prime}_{\phi b})\end{matrix}$

How $(t+9qs+9qZ^{\prime}_{\phi b}+9sZ^{\prime}_{\phi a})\in\mathbb{Z}$ then

$\displaystyle\begin{matrix}\vspace{0.2cm}v=9(t+9qs+9qZ^{\prime}_{\phi
b}+9sZ^{\prime}_{\phi a})+(t+9qs+9qZ^{\prime}_{\phi b}+9sZ^{\prime}_{\phi
a})\rightarrow v\equiv(Z^{\prime}_{\phi a}*Z^{\prime}_{\phi
b})\leavevmode\nobreak\ (mod\leavevmode\nobreak\ 9)\\\
\vspace{0.2cm}\therefore\\\ \vspace{0.2cm}v\in(Z^{\prime}_{\phi
a}*Z^{\prime}_{\phi b})\end{matrix}$

####  9.1.2 Associative property for the sum

Let

$\displaystyle\phi^{a},\phi^{b},\phi^{c}\in\zeta_{\phi}\rightarrow[\phi^{a}+\phi^{b}]+\phi^{c}=\phi^{a}+[\phi^{b}+\phi^{c}]$

Let’s supposed

$\displaystyle\begin{matrix}Z_{\phi a}\in\phi^{a},Z_{\phi
b}\in\phi^{b},Z_{\phi c}\in\phi^{c}\end{matrix}$

Then

$\displaystyle\begin{matrix}\vspace{0.2cm}[\phi^{a}+\phi^{b}]+\phi^{c}=(Z_{\phi
a}+Z_{\phi b})+\phi^{c}\\\ \vspace{0.2cm}=[(Z_{\phi a}+Z_{\phi b}+Z_{\phi
c})^{a+b+c}]\\\ \vspace{0.2cm}=[\phi^{a}+(Z_{\phi b}+Z_{\phi c})^{b+c}]\\\
\vspace{0.2cm}=\phi^{a}+[Z_{\phi b}+Z_{\phi c}]^{b+c}\\\
\vspace{0.2cm}=\phi^{a}+[\phi^{b}+\phi^{c}]\end{matrix}$

###  9.2 Property of existence of a neutral element

$\phi^{\pm 9}\in\left\\{\zeta_{\phi}\right\\}$ is the neutral element.

####  9.2.1 Proof:

$\displaystyle\phi^{a}+\phi^{\pm 9}=\left\\{v\in\mathbb{Z}\mid v\equiv(Z_{\phi
a}+9)\leavevmode\nobreak\ (mod\hskip 2.84544pt9)\right\\}$

Then

$\displaystyle\begin{matrix}\vspace{0.2cm}v-(Z_{\phi
a}+9)=9t,t\in\mathbb{Z}\\\ \vspace{0.2cm}v=9t+(Z_{\phi a}+9)\\\
\vspace{0.2cm}v=9(t+1)+Z_{\phi a}\rightarrow v\equiv Z_{\phi
a}\leavevmode\nobreak\ (mod\hskip 2.84544pt9)\therefore
v\in(\phi\circledcirc\phi)^{a+9}\end{matrix}$

##  10 Group $\left\\{\zeta_{\phi}\right\\}+$

Proposition of elements

###  10.1 Opposite element

$\displaystyle\phi^{-a}\in\left\\{\zeta_{\phi}\right\\}$

is the opposite element

####  10.1.1 Proof:

$\displaystyle(-Z_{\phi a})\in\phi^{(-a)}$

Then by definition we can affirm that:

$\displaystyle\begin{matrix}\vspace{0.2cm}\phi^{a}+\phi^{(-a)}=\left\\{v\in\mathbb{Z}\mid
v\equiv[Z_{\phi a}+(-Z_{\phi a})\leavevmode\nobreak\ (mod\hskip
2.84544pt9)]\right\\}\\\ \vspace{0.2cm}v=9t+[Z_{\phi a}+(-Z_{\phi
a})],t\in\mathbb{Z}\\\ \vspace{0.2cm}v=9t+0\rightarrow v\equiv 0(mod\hskip
2.84544pt9)\rightarrow v\equiv 9\leavevmode\nobreak\ (mod\hskip
2.84544pt9)\end{matrix}$

Ergo

$\displaystyle v\in\phi^{9}$

Finally

$\displaystyle\phi^{a}+\phi^{(-a)}=\phi^{\pm 9}$

###  10.2 Cancellation law of the sum

Let

$\displaystyle\phi^{a},\phi^{b},\phi^{c}\in\zeta_{\phi}\leavevmode\nobreak\
/\leavevmode\nobreak\
\phi^{a}+\phi^{c}=\phi^{b}+\phi^{c}\rightarrow\phi^{a}=\phi^{b}$

####  10.2.1 Proof:

As we know by definition:

$\displaystyle(\phi+\phi)^{a+c}=(\phi+\phi)^{b+c}=\left\\{v\in\mathbb{Z}\mid
v\equiv Z_{\phi a}+Z_{\phi c}=Z_{\phi b}+Z_{\phi c}\leavevmode\nobreak\
(mod\hskip 2.84544pt9)\right\\}$

From where:

$\displaystyle\begin{matrix}\vspace{0.2cm}v-[(Z_{\phi a}+Z_{\phi c})=(Z_{\phi
b}+Z_{\phi c})]=9t,t\in\mathbb{Z}\\\ \vspace{0.2cm}v=9t+[(Z_{\phi a}+Z_{\phi
c})=(Z_{\phi b}+Z_{\phi c})]\\\ v=9t+[(Z_{\phi a}+Z_{\phi c})+(-Z_{\phi
c})=(Z_{\phi b}+Z_{\phi c})+(-Z_{\phi c})]\\\ \vspace{0.2cm}v=9t+[(Z_{\phi
a}+Z_{\phi c}-Z_{\phi c})=(Z_{\phi b}+Z_{\phi c}-Z_{\phi c})]\\\
\vspace{0.2cm}v=9t+[Z_{\phi a}=Z_{\phi b}]\rightarrow v\equiv Z_{\phi
a}=Z_{\phi b}\leavevmode\nobreak\ (mod\hskip 2.84544pt9)\therefore
v\in\phi^{(a=b)}\end{matrix}$

##  11 Conjecture Primal

As we mentioned before, the Primal conjecture is a corollary of the "Acuña
Theorem" in chapter 8, and goes in this way:

$\displaystyle\phi^{a}\odot\phi^{b}\equiv\phi^{c}\leavevmode\nobreak\
(mod\leavevmode\nobreak\ 9)\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \exists\leavevmode\nobreak\
Z_{a},\leavevmode\nobreak\ Z_{b}\leavevmode\nobreak\ /\leavevmode\nobreak\
Z_{a}\odot Z_{b}=Z_{c}$

This conjecture says, that any operations have a class or primal number as
result, that means we have a solution that is congruent mod 9, but ¿if the
congruence exist that means all member of the class are solutions?, or ¿One or
more numbers are solution to that problem?.
In the case that the resulting class is congruent but not all its components
are results, we can deduce that at least there are numbers within these
classes that satisfy all the obtained congruence.

$\displaystyle\phi^{a}\odot\phi^{b}\equiv\phi^{c}\leavevmode\nobreak\
(mod\leavevmode\nobreak\ 9)\leavevmode\nobreak\
\Rightarrow\leavevmode\nobreak\ \neg\forall\leavevmode\nobreak\
Z_{a},\leavevmode\nobreak\ Z_{b}\leavevmode\nobreak\ /\leavevmode\nobreak\
Z_{a}\odot Z_{b}=Z_{c}$

##  12 Arithmetic Primal

All the early theorems are the base for the arithmetic tables presented in the
paper, Acuña and Marrero (2021)[5] So we gonna show again and add some
properties and new tables.

###  12.1 Primal additive Table

$\displaystyle\forall\leavevmode\nobreak\ \phi^{a}+\phi^{b}=\phi^{c}$

+ | $\phi^{1}$ | $\phi^{2}$ | $\phi^{3}$ | $\phi^{4}$ | $\phi^{5}$ | $\phi^{6}$ | $\phi^{7}$ | $\phi^{8}$ | $\phi^{9}$
---|---|---|---|---|---|---|---|---|---
$\phi^{1}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$
$\phi^{2}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$
$\phi^{3}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$
$\phi^{4}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$
$\phi^{5}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$
$\phi^{6}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$
$\phi^{7}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$
$\phi^{8}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$
$\phi^{9}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$

Table 4.

In this table, the cycle of infinite repetitions of the primal values can be
clearly appreciated.

###  12.2 Primal subtraction table

$\displaystyle\forall\leavevmode\nobreak\ \phi^{a}-\phi^{b}=\phi^{c}$

Like the subtraction is not commutative we have two tables, the first to
positive results and the second for the negative results.

- | $\phi^{1}$ | $\phi^{2}$ | $\phi^{3}$ | $\phi^{4}$ | $\phi^{5}$ | $\phi^{6}$ | $\phi^{7}$ | $\phi^{8}$ | $\phi^{9}$
---|---|---|---|---|---|---|---|---|---
$\phi^{-1}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$
$\phi^{-2}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$
$\phi^{-3}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$
$\phi^{-4}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$
$\phi^{-5}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$
$\phi^{-6}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$
$\phi^{-7}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$ | $2_{\phi}$
$\phi^{-8}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$ | $1_{\phi}$
$\phi^{-9}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$

Table 5.

+ | $\phi^{-1}$ | $\phi^{-2}$ | $\phi^{-3}$ | $\phi^{-4}$ | $\phi^{-5}$ | $\phi^{-6}$ | $\phi^{-7}$ | $\phi^{-8}$ | $\phi^{-9}$
---|---|---|---|---|---|---|---|---|---
$\phi^{1}$ | $-9_{\phi}$ | $-8_{\phi}$ | $-7_{\phi}$ | $-6_{\phi}$ | $-5_{\phi}$ | $-4_{\phi}$ | $-3_{\phi}$ | $-2_{\phi}$ | $-1_{\phi}$
$\phi^{2}$ | $-1_{\phi}$ | $-9_{\phi}$ | $-8_{\phi}$ | $-7_{\phi}$ | $-6_{\phi}$ | $-5_{\phi}$ | $-4_{\phi}$ | $-3_{\phi}$ | $-2_{\phi}$
$\phi^{3}$ | $-2_{\phi}$ | $-1_{\phi}$ | $-9_{\phi}$ | $-8_{\phi}$ | $-7_{\phi}$ | $-6_{\phi}$ | $-5_{\phi}$ | $-4_{\phi}$ | $-3_{\phi}$
$\phi^{4}$ | $-3_{\phi}$ | $-2_{\phi}$ | $-1_{\phi}$ | $-9_{\phi}$ | $-8_{\phi}$ | $-7_{\phi}$ | $-6_{\phi}$ | $-5_{\phi}$ | $-4_{\phi}$
$\phi^{5}$ | $-4_{\phi}$ | $-3_{\phi}$ | $-2_{\phi}$ | $-1_{\phi}$ | $-9_{\phi}$ | $-8_{\phi}$ | $-7_{\phi}$ | $-6_{\phi}$ | $-5_{\phi}$
$\phi^{6}$ | $-5_{\phi}$ | $-4_{\phi}$ | $-3_{\phi}$ | $-2_{\phi}$ | $-1_{\phi}$ | $-9_{\phi}$ | $-8_{\phi}$ | $-7_{\phi}$ | $-6_{\phi}$
$\phi^{7}$ | $-6_{\phi}$ | $-5_{\phi}$ | $-4_{\phi}$ | $-3_{\phi}$ | $-2_{\phi}$ | $-1_{\phi}$ | $-9_{\phi}$ | $-8_{\phi}$ | $-7_{\phi}$
$\phi^{8}$ | $-7_{\phi}$ | $-6_{\phi}$ | $-5_{\phi}$ | $-4_{\phi}$ | $-3_{\phi}$ | $-2_{\phi}$ | $-1_{\phi}$ | $-9_{\phi}$ | $-8_{\phi}$
$\phi^{9}$ | $-8_{\phi}$ | $-7_{\phi}$ | $-6_{\phi}$ | $-5_{\phi}$ | $-4_{\phi}$ | $-3_{\phi}$ | $-2_{\phi}$ | $-1_{\phi}$ | $-9_{\phi}$

Table 6.

Also we can obtain 0 when we subtract the same class, if and only if the
number they are representing is the same.

$\displaystyle\begin{matrix}\phi^{a}-\phi^{b}=0\equiv v_{a}-v_{b}=0\mid
v_{a}=a_{b}\rightarrow\phi^{a}=\phi^{b}\end{matrix}$

These two tables in particular are complex and less intuitive, since it is an
operation that depends on the size of the number and the signs to determine if
the result will be positive or negative, but it also affects the result in a
modular way.

###  12.3 Primal Division Table

$\div$ | $\phi^{1}$ | $\phi^{2}$ | $\phi^{3}$ | $\phi^{4}$ | $\phi^{5}$ | $\phi^{6}$ | $\phi^{7}$ | $\phi^{8}$ | $\phi^{9}$
---|---|---|---|---|---|---|---|---|---
$\phi^{1}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$
$\phi^{2}$ | $5_{\phi}$ | $1_{\phi}$ | $6_{\phi}$ | $2_{\phi}$ | $7_{\phi}$ | $3_{\phi}$ | $8_{\phi}$ | $7_{\phi}$ | $9_{\phi}$
$\phi^{3}$ | $\varnothing$ | $\varnothing$ | $4_{\phi};1_{\phi};7_{\phi}$ | $\varnothing$ | $\varnothing$ | $5_{\phi};2_{\phi};8_{\phi}$ | $\varnothing$ | $\varnothing$ | $3_{\phi};9_{\phi};6_{\phi}$
$\phi^{4}$ | $7_{\phi}$ | $5_{\phi}$ | $3_{\phi}$ | $1_{\phi}$ | $8_{\phi}$ | $6_{\phi}$ | $4_{\phi}$ | $2_{\phi}$ | $9_{\phi}$
$\phi^{5}$ | $2_{\phi}$ | $4_{\phi}$ | $6_{\phi}$ | $8_{\phi}$ | $1_{\phi}$ | $3_{\phi}$ | $5_{\phi}$ | $7_{\phi}$ | $9_{\phi}$
$\phi^{6}$ | $\varnothing$ | $\varnothing$ | $5_{\phi};2_{\phi};8_{\phi}$ | $\varnothing$ | $\varnothing$ | $4_{\phi};1_{\phi};7_{\phi}$ | $\varnothing$ | $\varnothing$ | $3_{\phi};9_{\phi};6_{\phi}$
$\phi^{7}$ | $4_{\phi}$ | $8_{\phi}$ | $3_{\phi}$ | $7_{\phi}$ | $2_{\phi}$ | $6_{\phi}$ | $1_{\phi}$ | $5_{\phi}$ | $9_{\phi}$
$\phi^{8}$ | $8_{\phi}$ | $7_{\phi}$ | $6_{\phi}$ | $5_{\phi}$ | $4_{\phi}$ | $3_{\phi}$ | $2_{\phi}$ | $1_{\phi}$ | $9_{\phi}$
$\phi^{9}$ | $\varnothing$ | $\varnothing$ | $\varnothing$ | $\varnothing$ | $\varnothing$ | $\varnothing$ | $\varnothing$ | $\varnothing$ | $Z_{\phi}$

Table 7.

Also we know by the Acuña’s Theorem, That:

$\displaystyle\forall\hskip 2.84544pt\leavevmode\nobreak\
\frac{\phi^{a}}{\phi^{b}}=\phi^{c}\rightarrow\frac{\phi^{a}}{\phi^{c}}=\phi^{b}\rightarrow\phi^{b}*\phi^{c}=\phi^{a}$

###  12.4 Primal multiplication table

$\displaystyle\forall\leavevmode\nobreak\ \phi^{a}*\phi^{b}=\phi^{c}$

* | $\phi^{1}$ | $\phi^{2}$ | $\phi^{3}$ | $\phi^{4}$ | $\phi^{5}$ | $\phi^{6}$ | $\phi^{7}$ | $\phi^{8}$ | $\phi^{9}$
---|---|---|---|---|---|---|---|---|---
$\phi^{1}$ | $1_{\phi}$ | $2_{\phi}$ | $3_{\phi}$ | $4_{\phi}$ | $5_{\phi}$ | $6_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $9_{\phi}$
$\phi^{2}$ | $2_{\phi}$ | $4_{\phi}$ | $6_{\phi}$ | $8_{\phi}$ | $1_{\phi}$ | $3_{\phi}$ | $5_{\phi}$ | $7_{\phi}$ | $9_{\phi}$
$\phi^{3}$ | $3_{\phi}$ | $6_{\phi}$ | $9_{\phi}$ | $3_{\phi}$ | $6_{\phi}$ | $9_{\phi}$ | $3_{\phi}$ | $6_{\phi}$ | $9_{\phi}$
$\phi^{4}$ | $4_{\phi}$ | $8_{\phi}$ | $3_{\phi}$ | $7_{\phi}$ | $2_{\phi}$ | $6_{\phi}$ | $1_{\phi}$ | $5_{\phi}$ | $9_{\phi}$
$\phi^{5}$ | $5_{\phi}$ | $1_{\phi}$ | $6_{\phi}$ | $2_{\phi}$ | $7_{\phi}$ | $3_{\phi}$ | $8_{\phi}$ | $4_{\phi}$ | $9_{\phi}$
$\phi^{6}$ | $6_{\phi}$ | $3_{\phi}$ | $9_{\phi}$ | $6_{\phi}$ | $3_{\phi}$ | $9_{\phi}$ | $6_{\phi}$ | $3_{\phi}$ | $9_{\phi}$
$\phi^{7}$ | $7_{\phi}$ | $5_{\phi}$ | $3_{\phi}$ | $1_{\phi}$ | $8_{\phi}$ | $6_{\phi}$ | $4_{\phi}$ | $2_{\phi}$ | $9_{\phi}$
$\phi^{8}$ | $8_{\phi}$ | $7_{\phi}$ | $6_{\phi}$ | $5_{\phi}$ | $4_{\phi}$ | $3_{\phi}$ | $2_{\phi}$ | $1_{\phi}$ | $9_{\phi}$
$\phi^{9}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$

Table 8.

###  12.5 Primal power table

$()^{*}$ | $()^{2}$ | $()^{3}$ | $()^{4}$ | $()^{5}$ | $()^{6}$ | $()^{7}$ | $()^{8}$ | $()^{9}$ | $()^{10}$ | $()^{11}$ | $()^{12}$ | $()^{13}$ | $()^{14}$ | $()^{15}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\phi^{1}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$ | $1_{\phi}$
$\phi^{2}$ | $4_{\phi}$ | $8_{\phi}$ | $7_{\phi}$ | $5_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $4_{\phi}$ | $8_{\phi}$ | $7_{\phi}$ | $5_{\phi}$ | $1_{\phi}$ | $2_{\phi}$ | $4_{\phi}$ | $8_{\phi}$
$\phi^{3}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$
$\phi^{4}$ | $7_{\phi}$ | $1_{\phi}$ | $4_{\phi}$ | $7_{\phi}$ | $1_{\phi}$ | $4_{\phi}$ | $7_{\phi}$ | $1_{\phi}$ | $4_{\phi}$ | $7_{\phi}$ | $1_{\phi}$ | $4_{\phi}$ | $7_{\phi}$ | $1_{\phi}$
$\phi^{5}$ | $7_{\phi}$ | $8_{\phi}$ | $4_{\phi}$ | $2_{\phi}$ | $1_{\phi}$ | $5_{\phi}$ | $7_{\phi}$ | $8_{\phi}$ | $4_{\phi}$ | $2_{\phi}$ | $1_{\phi}$ | $5_{\phi}$ | $7_{\phi}$ | $8_{\phi}$
$\phi^{6}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9{\phi}$ | $9_{\phi}$ | $9_{\phi}$
$\phi^{7}$ | $4_{\phi}$ | $1_{\phi}$ | $7_{\phi}$ | $4_{\phi}$ | $1_{\phi}$ | $7_{\phi}$ | $4_{\phi}$ | $1_{\phi}$ | $7_{\phi}$ | $4_{\phi}$ | $1_{\phi}$ | $7_{\phi}$ | $4_{\phi}$ | $1_{\phi}$
$\phi^{8}$ | $1_{\phi}$ | $8_{\phi}$ | $1_{\phi}$ | $8{\phi}$ | $1_{\phi}$ | $8_{\phi}$ | $1_{\phi}$ | $8_{\phi}$ | $1_{\phi}$ | $8_{\phi}$ | $1_{\phi}$ | $8_{\phi}$ | $1_{\phi}$ | $8_{\phi}$
$\phi^{9}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$ | $9_{\phi}$

Table 9.

In this table we can see all the results for the powers from the square to the
power 15, looks like a cycle appears when we reach $(\phi^{Z})^{8}$ and reboot
in $(\phi^{Z})^{14}$, so this is a open problem.

####  12.5.1 Proposal Conjecture 2:"infinite cycle of powers"

$\displaystyle\forall\hskip 2.84544pt(\phi^{Z})^{n}\hskip
2.84544pt\exists\hskip
2.84544pt(\phi^{Z})^{n+6}\mid(\phi^{Z})^{n}\equiv(\phi^{Z})^{n+6}$

This means that there is also an infinite cycle for the powers, which is
symmetric and constant, this cycle would demonstrate the existence of
congruence in the powers, allowing us to group polynomial equations so that
they have the same modular congruence or in this case the same primal class.

##  13 Corollary of Problems

All this problem are in (mod 9)

  * •

$\displaystyle\textquestiondown X^{3}+Y^{3}+Z^{3}=K\equiv
X^{9}+Y^{9}+Z^{9}=K?$

  * •

$\displaystyle\textquestiondown(Z_{\phi a})^{2}+(Z_{\phi
b})^{3}=k\leavevmode\nobreak\ for\leavevmode\nobreak\ any\leavevmode\nobreak\
specific\leavevmode\nobreak\ k$

  * •

$\displaystyle Z^{n}+X^{n}=K^{n}\equiv Z^{n+6}+X^{n+6}=K$

## References

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  * Tarazona et al. [2021] E. J. A. Tarazona, S. Flores, and P. F. Marrero.  Matrize_$\phi$^ 9. clasificación de números enteros. algebra primordial.  _Revista MATUA ISSN: 2389-7422_ , 8(1):10–45, 2021.
  * Vinogradov and Bernardo [1977] I. M. Vinogradov and E. A. Bernardo.  _Fundamentos de la teoría de los números_.  Mir, 1977.