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Abstract and Figures

According to a new mathematical definition, whole numbers are divided into two sets, one of which is the merger of the sequence of prime numbers and numbers zero and one. Three other definitions, deduced from this first, subdivide the set of whole numbers into four classes of numbers with own and unique arithmetic properties. The geometric distribution of these different types of whole numbers, in various closed matrices, is organized into exact value ratios to 3/2 or 1/1. Selon une nouvelle définition mathématique, les nombres entiers naturels se divisent en deux ensembles dont l'un est la fusion de la suite des nombres premiers et des nombres zéro et un. Trois autres définitions, déduites de cette première, subdivisent l'ensemble des nombres entiers naturels en quatre classes de nombres aux propriétés arithmétiques propres et uniques. La distribution géométrique de ces différents types d'entiers naturels, dans de diverses matrices fermées, s'organise en ratios exacts de valeur 3/2 ou 1/1.
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The Ultimate Numbers
The Ultimate Numbers and the 3/2 Ratio
Les nombres ultimes et le ratio 3/2
Jean-Yves BOULAY
Abstract. According to a new mathematical definition, whole numbers are divided into two sets, one of which is the merger of
the sequence of prime numbers and numbers zero and one. Three other definitions, deduced from this first, subdivide the set of
whole numbers into four classes of numbers with own and unique arithmetic properties. The geometric distribution of these
different types of whole numbers, in various closed matrices, is organized into exact value ratios to 3/2 or 1/1.
AMS subject classification: 11A41-11R29-11R21-11B39-11C20
Keywords: prime numbers, whole numbers, Sophie Germain numbers, Symmetry.
Résumé. Selon une nouvelle définition mathématique, les nombres entiers naturels se divisent en deux ensembles dont l’un est
la fusion de la suite des nombres premiers et des nombres zéro et un. Trois autres définitions, déduites de cette première,
subdivisent l’ensemble des nombres entiers naturels en quatre classes de nombres aux propriétés arithmétiques propres et
uniques. La distribution géométrique de ces différents types d’entiers naturels, dans de diverses matrices fermées, s’organise
en ratios exacts de valeur 3/2 ou 1/1.
1. Introduction
This study invests the whole numbers* set (ℕ) and proposes a mathematical definition to integrate the number zero (0) and the
number one (1) into the thus called prime numbers sequence. This set is called the set of ultimate numbers. The study of many
matrices of numbers such as, for example, the table of cross additions of the ten digit-numbers (from 0 to 9) highlights a non-
random arithmetic and geographic organization of these ultimate numbers. It also appears that this distinction between ultimate
and non-ultimate numbers (like also other proposed distinctions of different classes of whole numbers) is intimately linked to
the decimal system, in particular and mainly by an almost systematic opposition of the entities in a ratio to 3/2. Indeed this
ratio can only manifest itself in the presence of multiples of five (10/2) entities. Also, it is within matrices of ten times ten
numbers that the majority of demonstrations validating an opposition of entities in ratios to value 3/2 or /and value to 1/1 are
made.
* In statements, when this is not specified, the term "number" always implies a "whole number". Also, It is agreed that the
number zero (0) is well integrated into the set of whole numbers.
2. The ultimate numbers
2.1 Definition of an ultimate number
Considering the set of whole numbers, these are organized into two sets: ultimate numbers and non-ultimate numbers.
Ultimate numbers definition:
An ultimate number not admits any non-trivial divisor (whole number) being less than it.
Non-ultimate numbers definition:
A non-ultimate number admits at least one non-trivial divisor (whole number) being less than it.
Note: a non-trivial divisor of a whole number n is a whole number which is a divisor of n but distinct from n and from 1
(which are its trivial divisors).
2.2. The first ten ultimate numbers and the first ten non-ultimate numbers
Considering the previous double definition, the sequence of ultimate numbers is initialized by these ten numbers:
0
1
2
3
5
7
11
13
17
19
Considering the previous double definition, the sequence of non-ultimate numbers is initialized by these ten numbers:
4
6
8
9
10
12
14
15
16
18
2.3 Development
2.3.1 Other definitions
Let n be a whole number (belonging to ), this one is ultimate if no divisor (whole number) lower than its value and other than
1 divides it.
Let n be a whole number (belonging to ), this one is non-ultimate if at least one divisor (whole number) lower than its value
and other than 1 divides it.
2.3.2 Development
Below are listed, to illustration of definition, some of the first ultimate or non-ultimate numbers defined above, especially
particular numbers zero (0) and one (1).
- 0 is ultimate: although it admits an infinite number of divisors superior to it, since it is the first whole number, the
number 0 does not admit any divisor being inferior to it.
- 1 is ultimate: since the division by 0 has no defined result, the number 1 does not admit any divisor (whole number)
being less than it.
- 2 is ultimate: since the division by 0 has no defined result, the number 2 does not admit any divisor* being less than
it.
- 4 is non-ultimate: the number 4 admits the number 2 (number being less than it) as divisor*.
- 6 is non-ultimate: the number 6 admits numbers 2 and 3 (numbers being less than it) as divisors*.
- 7 is ultimate: since the division by 0 has no defined result, the number 7 does not admit any divisor* being less than
it. The non-trivial divisors 2, 3, 4, 5 and 6 cannot divide it into whole numbers.
- 12 is non-ultimate: the number 6 admits numbers 2, 3, 4 and 6 (numbers being less than it) as divisors*.
Thus, by these previous definitions, the set of whole numbers is organized into these two entities:
- the set of ultimate numbers, which is the fusion of the prime numbers sequence with the numbers 0 and 1.
- the set of non-ultimate numbers identifying to the non-prime numbers sequence, deduced from the numbers 0 and 1.
* non-trivial divisor.
2.4 Conventional designations
As "primes" designates prime numbers, it is agree that designation "ultimates" designates ultimate numbers. Also it is agree
that designation "non-ultimates" designates non-ultimate numbers. Other conventional designations will be applied to the
different classes or types of whole numbers later introduced.
2.5 The ultimate numbers and the decimal system
It turns out that the tenth ultimate number is the number 19, a number located in twentieth place in the sequence of the whole
numbers. This peculiarity undeniably links the ultimate numbers and the decimal system. So the first twenty numbers (twice
ten numbers) are organized into different 1/1 and 3/2 ratios according to their different attributes.
By the nature of the decimal system, as shown in Figure 1, the ten digit numbers (digits confused as numbers) are opposed to
the first ten non-digit numbers by a ratio of 1/1. Also, there are exactly the same quantity of ultimates and non-ultimates among
these twenty numbers, so ten entities in each category. In a double 3/2 value ratio, six ultimates versus four are among the ten
digit numbers and six non-ultimates versus four are among the first ten non-digit numbers.
10 digit numbers
← 1/1 ratio →
10 non-digit numbers
6 ultimates
← 3/2 ratio →
4 ultimates
1
3
4
5
7
9
10
13
14
16
4 non-ultimates
← 2/3 ratio
6 non-ultimates
3/2 ratio
2/3 ratio
Fig.1 Differentiation of the 20 fundamental numbers according to their digitality or non-digitality: the 10 digit numbers (digits confused as
numbers) and the first 10 non-digit numbers. Différenciation des 20 nombres fondamentaux selon leur digitalité ou non digitalité : les 10
chiffres nombres (chiffres confondus comme nombres) et les 10 premiers nombres non digitaux.
As shown in Figure 2, it is also possible to describe this arithmetic phenomenon by crossing criteria. Thus, the first ten
ultimates are opposed to the ten non-ultimates by a 1/1 value ratio. Also, there are exactly the same quantity of digit numbers
and non-digit numbers among these twenty numbers. In a twice 3/2 ratio, six digits versus four are among the ten ultimates and
six non-digits numbers versus four are among the first ten non-ultimates.
10 first ultimate numbers:
← 1/1 ratio
10 first non-ultimate numbers:
6 digit numbers
← 3/2 ratio
4 digit numbers
1
3
5
7
13
19
4
9
10
14
4 non-digit numbers
← 2/3 ratio
6 non-digit numbers
3/2 ratio
2/3 ratio
Fig.2 Differentiation of the 20 fundamental numbers according to their ultimity or non-ultimity: 10 ultimates versus 10 non-ultimates.
Différenciation des 20 nombres fondamentaux selon leur ultimité ou non ultimité : 10 ultimes contre 10 non ultimes.
Technical remark: due to a certain complexity of the phenomena presented and to clarify their understanding, no figure (table)
has a title but just a legend in this paper.
2.6 The twenty fundamental numbers
Whole numbers sequence is therefore initialized by twenty numbers with symmetrically and asymmetrically complementary
characteristics of reversible 1/1 and 3/2 ratios. This transcendent entanglement of the first twenty numbers according to their
ultimate or non-ultimate nature (ultimate numbers or non-ultimate numbers) and according to their digit or non-digit nature
(digits or non-digit numbers) allows, by convention, to qualify them as "fundamental numbers" among the whole numbers set.
Figure 3 describes the total entanglement of these twenty fundamental numbers.
digitality
ultimity
0
1
2
3
5
7
4
6
8
9
non-ultimity
11
13
17
19
10
12
14
15
16
18
non-digitality
Fig. 3 Entanglement of the 20 fundamental numbers according to their ultimity or non-ultimity and their digitality or non-
digitality. Intrication des 20 nombres fondamentaux selon leur ultimité ou non ultimité et leur digitalité ou non digitalité.
Thus, the set of the first twenty whole numbers is simultaneously made up to a set of twenty entities including ten ultimate
numbers and ten non-ultimate numbers and to a (same) set of twenty entities including ten digit numbers (10 digits ) and ten
non-digit numbers (not digits). Also, each of these four entangled subsets of ten entities with their own properties opposing two
by two in 1/1 value ratio is composed of two opposing subsets in 3/2 value ratio according to the mixed properties of its
components. This set of the first twenty numbers is defined as the set of fundamental numbers among the whole numbers.
So it is agree that designation "fundamentals" designates these twenty fundamental numbers previously defined.
2.7 The thirty initial numbers
Also, according to the progressive consideration of three sets of 10, 20 and then 30 entities (the first thirty whole numbers), the
ratio between the ultimate and non-ultimate numbers increases from 3/2 (10 numbers) to 1/1 ( 20 numbers) then switches to
2/3 (30 numbers). Thus (Figure 4), depending on whether we consider the first ten, the first twenty and then the first thirty
whole numbers, 6 ultimates are opposed to 4 non-ultimates, then 10 ultimates are opposed to 10 non-ultimates then finally 12
ultimates are opposed to 18 non-ultimates. Beyond this triple set, no similar organization of (consecutive) groups of ten entities
takes place. These thirty numbers are therefore here called "initials" among the set of natural numbers.
30 initial numbers
20 fundamental numbers
10 digits (digit numbers)
0 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
6 ultimates / 4 non-ultimates: 3/2 ratio
10 ultimates / 10 non-ultimates: 1/1 ratio
12 ultimates / 18 non-ultimates: 3/2 ratio
Fig. 4 Switching from the 3/2 ratio to the 2/3 ratio according to the classification of the first thirty whole numbers and their degree of ultimity.
Basculement du ratio 3/2 vers le ratio 2/3 selon le classement des trente premiers nombres entiers naturels et leur degré d’ultimité.
3. Addition matrix of the ten digits
The table in Figure 5 represents the matrix of the hundred different possible sums of additions (crossed) of the ten digit
numbers (from 0 to 9) of the decimal system (ie the first ten whole numbers). Within this table operate multiple singular
arithmetic phenomena depending on the ultimate or non-ultimate nature of the values of these hundred sums and their
geographic distribution including mainly various 3/2 value ratios often transcendent.
3.1 Sixty versus forty numbers: 3/2 ratio
Among these hundred values, there are 40 ultimate numbers (5x x = 8) and consecutively 60 non-ultimate numbers (5y y
= 12). These two sets therefore oppose each other in an exact 2/3 value ratio.
+
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10
2
2
3
4
5
6
7
8
9
10
11
3
3
4
5
6
7
8
9
10
11
12
4
4
5
6
7
8
9
10
11
12
13
5
5
6
7
8
9
10
11
12
13
14
6
6
7
8
9
10
11
12
13
14
15
7
7
8
9
10
11
12
13
14
15
16
8
8
9
10
11
12
13
14
15
16
17
9
9
10
11
12
13
14
15
16
17
18
40 ultimates
6
5 + 5
4 + 4 + 4
3 + 3 + 3 + 3
1(n)
2(n-1)
3(n-2)
4(n-3)
60 non-ultimates
4
5 + 5
6 + 6 + 6
7 + 7 + 7 + 7
(2n/3)
2((2n/3) + 1)
3((2n/3)+2)
4((2n/3) + 3)
Fig. 5 Cross additions table of the ten digit numbers. Tableau d’additions croisées des dix chiffres nombres.
Also in this table, the quantities of the values equal to an ultimate number decrease regularly from 6 entities (n) to 3 from the
first column to the tenth. This decrease is distinguished by a double arithmetic phenomenon: the first column, which represents
the additions of the ten digit numbers with the first of these (0), therefore total a unique value number of 6 ultimates (n); the
next two columns add up to two same ultimates quantities and that value (5) is just one unit less than the number in the first
addition column; the following three columns total three same values (4) lower by one unit than the two preceding columns
then finally, the four final columns continue and close this regular arithmetic arrangement with four same values of non-
ultimate numbers (3) also lower by one unit to the three preceding columns. The same arithmetic arrangement is observed for
the counting of sums equal to a non-ultimate number but in an increasing direction of the quantities of the non-ultimate
numbers counted and with a source number (4) equal to 2n/3. By the nature of this crosstab, the same phenomenon naturally
occurs from line to line.
In this matrix, the addition columns are therefore grouped by one, two, three and then four arithmetic entities. Also, from the
value n (6 ultimates in first column of additions), the complet sum of ultimates is obtained by this formula:
n + 2(n - 1) + 3(n - 2) + 4(n - 3)
The complet sum of non-ultimates is obtained by this formula:
(2n/3) + 2((2n/3) + 1) + 3((2n/3) + 2) + 4((2n/3) + 3)
This phenomenon is directly related to the decimal system organized from ten entities: the value 10 is indeed equal to the sum
of four progressive values: 1 + 2 + 3 + 4 = 10.
3.2 Twenty-four versus sixteen ultimates: 3/2 ratio
Among the 50 sums equal to the addition of the 10 digit numbers (from 0 to 9) with the first 5 digits (from 0 to 4), there are 24
ultimates and among the 50 sums equal to the addition of the 10 digit numbers (from 0 to 9) with the last 5 digits (from 5 to 9),
there are 16 ultimates. These two groups are therefore in opposition (Figure 6) in a ratio of 3/2.
Almong the first 50 values:
24 ultimates
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
11
3
4
5
6
7
8
9
10
11
12
4
5
6
7
8
9
10
11
12
13
5
6
7
8
9
10
11
12
13
14
6
7
8
9
10
11
12
13
14
15
7
8
9
10
11
12
13
14
15
16
8
9
10
11
12
13
14
15
16
17
9
10
11
12
13
14
15
16
17
18
Almong the last 50 values:
16 ultimates
Fig. 6 Cross addition semi tables of the 10 digits generating a 3/2 value ratio on the distribution of the ultimate numbers. Semi
tableaux d’additions croisées des 10 chiffres nombres générant un ratio de valeur 3/2 de la distribution des nombres ultimes.
Among the 40 sums equal to an ultimate number, 24 correspond (Figure 7) to a digit of the decimal system (from 0 to 9) and
16 to a number greater than 9 (the last digit of the decimal system).
Among the digits:
24 ultimates
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
11
3
4
5
6
7
8
9
10
11
12
4
5
6
7
8
9
10
11
12
13
5
6
7
8
9
10
11
12
13
14
6
7
8
9
10
11
12
13
14
15
7
8
9
10
11
12
13
14
15
16
8
9
10
11
12
13
14
15
16
17
9
10
11
12
13
14
15
16
17
18
Among the non-digits:
16 ultimates
Fig. 7 Cross addition tables of the 10 digits generating a 3/2 value ratio on the distribution of the ultimate numbers depending on the
digital nature of values (digits or not digits). Tableau d’addition croisées des 10 chiffres nombres générant un ratio de valeur 3/2 de la
distribution des nombres ultimes selon la nature digitale des valeurs (chiffres nombres ou non chiffres nombres).
3.3 Configurations at 3/2 transcendent double ratio
By symmetrically splitting the addition matrix of the ten digits into two sub-matrices on 60 external entities versus 40 internal
entities, as presented in the left part of Figure 8, it appears that the non-ultimate numbers and the ultimate numbers always
oppose in different sets of 3/2 value ratios according to their identical or opposite natures. This is therefore verified both inside
the two sub-matrices and transversely to these two sub-matrices.
External sub-matrix
to 4 times 15 numbers
(60 sums)
3/2 ratio
Internal sub-matrix
to 4 times 10 numbers
(40 sums)
Internal sub-matrix
to 4 times 15 numbers
(60 sums)
3/2 ratio
External sub-matrix
to 4 times 10 numbers
(40 sums)
0
1
2
3
4
5
6
7
8
9
1
2
3
4
7
8
9
10
2
3
4
9
10
11
3
4
11
12
4
13
5
14
6
7
14
15
7
8
9
14
15
16
8
9
10
11
14
15
16
17
9
10
11
12
13
14
15
16
17
18
5
6
5
6
7
8
5
6
7
8
9
10
5
6
7
8
9
10
11
12
6
7
8
9
10
11
12
13
8
9
10
11
12
13
10
11
12
13
12
13
4
5
4
5
6
7
4
5
6
7
8
9
4
5
6
7
8
9
10
11
4
5
6
7
8
9
10
11
12
13
5
6
7
8
9
10
11
12
13
14
7
8
9
10
11
12
13
14
9
10
11
12
13
14
11
12
13
14
13
14
0
1
2
3
6
7
8
9
1
2
3
8
9
10
2
3
10
11
3
12
6
15
7
8
15
16
8
9
10
15
16
17
9
10
11
12
15
16
17
18
36 non-ultimates
24 ultimates
3/2 ratio
3/2 ratio
3/2 ratio
24 non-ultimates
16 ultimates
3/2 ratio
36 non-ultimates
24 ultimates
3/2 ratio
3/2 ratio
3/2 ratio
24 non-ultimates
16 ultimates
3/2 ratio
Fig.8 External and internal sub-matrices on 60 versus 40 numbers generating opposite sets of numbers in transcendent ratios of value 3/2 according to the
ultimity or not ultimity of their components. Sous matrices externes et internes de 60 contre 40 nombres générant des ensembles de nombres s’opposant
en ratios transcendants de valeur 3/2 selon l’ultimité ou non ultimité de leur composants.
Also, the same phenomena are observed by considering two sub-matrices on 60 internal entities versus 40 external entities as
presented in right part of Figure 8. Finally, the same phenomena still occur by considering two sub-matrices on 60
geographically located entities northwest and 40 geographically located entities southeast as shown in Figure 9.
Northwest sub-matrix to 4 times 15 numbers
(60 sums)
← 3/2 ratio →
Southeast sub-matrix to 4 times 10 numbers
(40 sums)
0
1
2
3
4
5
6
7
8
9
1
2
3
4
6
7
8
9
2
3
4
7
8
9
3
4
8
9
4
9
5
6
7
8
9
10
11
12
13
14
6
7
8
9
11
12
13
14
7
8
9
12
13
14
8
9
13
14
9
14
← 3/2 ratio →
5
10
5
6
10
11
5
6
7
10
11
12
5
6
7
8
10
11
12
13
10
15
10
11
15
16
10
11
12
15
16
17
10
11
12
13
15
16
17
18
36 non-ultimates
24 ultimates
3/2 ratio
3/2 ratio
3/2 ratio
24 non-ultimates
16 ultimates
3/2 ratio
Fig.9 Northwest and southeast sub-matrices on 60 versus 40 numbers generating opposite sets of numbers in
transcendent ratios of value 3/2 according to the ultimity or not ultimity of their components. Sous matrices nord-ouest
et sud-est de 60 contre 40 nombres générant des ensembles de nombres s’opposant en ratios transcendants de valeur 3/2
selon l’ultimité ou non ultimité de leur composants.
4. Addition matrix of the twenty fundamental numbers
The matrix of 100 numbers (Figure 10) of the additions of the ten digit numbers and of the following ten, that of the twenty
fundamental numbers, generates 70 non-ultimates (5x x = 14) and 30 ultimates (5y y = 6). These two categories of
numbers are not distributed randomly in this matrix but in singular arithmetic arrangements. Thus, the first two addition
columns each total 6 non-ultimates and 4 ultimates; the last two total 8 non-ultimate and 2 ultimate each. The six central
columns all have the same values of 7 and 3 numbers, respectively non-ultimate and ultimate. These six central columns
therefore oppose, in 3/2 ratios, their global quantity of non-ultimate and ultimate numbers to that of the four peripheral
columns with respectively 42 versus 28 non-ultimates and 18 versus 12 ultimates.
+
10
11
12
13
14
15
16
17
18
19
0
10
11
12
13
14
15
16
17
18
19
1
11
12
13
14
15
16
17
18
19
20
2
12
13
14
15
16
17
18
19
20
21
6 columns
(60 sums)
← 3/2 ratio →
4 columns
(40 sums)
3
13
14
15
16
17
18
19
20
21
22
4
14
15
16
17
18
19
20
21
22
23
42 non-ultimates
18 ultimates
← 3/2 ratio →
← 3/2 ratio →
28 non-ultimates
12 ultimates
5
15
16
17
18
19
20
21
22
23
24
6
16
17
18
19
20
21
22
23
24
25
7
17
18
19
20
21
22
23
24
25
26
8
18
19
20
21
22
23
24
25
26
27
9
19
20
21
22
23
24
25
26
27
28
70 non-ultimates
6
6
7
7
7
7
7
7
8
8
30 ultimates
4
4
3
3
3
3
3
3
2
2
42 non-ultimates
18 ultimates
28 non-ultimates
12 ultimates
Fig.10 Addition matrix of the twenty fundamental numbers. Matrice d’additions des vingt nombres fondamentaux.
4.1 Sub-matrices to sixty and forty numbers
In the addition matrix of the twenty fundamental numbers (to one hundred sums) two sub-matrices oppose, left part of Figure
11, their quantities of reciprocal non-ultimates and their quantities of reciprocal ultimates in ratios of 3/2 value. These sub-
matrices on 60 versus 40 numbers are themselves each composed of two sub-zones with the numbers of entities opposing in
3/2 ratios: sub-matrix on 36 + 24 entities and sub-matrix on 24 + 16 entities. This arithmetic arrangement is a geometric variant
of the remarkable identity (a + b)2 = a2 + 2ab + b2 where a and b here have as values 6 and 4, values opposing in the 3/2 ratio.
This remarkable identity will be more widely investigated in chapter 7.1 where very singular phenomena are presented.
Sub-matrix to 36 + 24
numbers (60 sums)
a2 + ab
3/2 ratio
Sub-matrix to 24 + 16
numbers (40 sums)
b2 + ba
Sub-matrix to 24 + 36
numbers (60 sums)
ab + a2
3/2 ratio
Sub-matrix to 16 + 24
numbers (40 sums)
ba + b2
12
13
14
15
16
17
13
14
15
16
17
18
14
15
16
17
18
19
15
16
17
18
19
20
14
15
16
17
18
19
20
21
22
23
15
16
17
18
19
20
21
22
23
24
16
17
24
25
17
18
25
26
18
19
26
27
19
20
27
28
10
11
18
19
11
12
19
20
12
13
20
21
13
14
21
22
18
19
20
21
22
23
19
20
21
22
23
24
20
21
22
23
24
25
21
22
23
24
25
26
10
11
18
19
11
12
19
20
12
13
20
21
13
14
21
22
14
15
16
17
18
19
20
21
22
23
15
16
17
18
19
20
21
22
23
24
18
19
20
21
22
23
19
20
21
22
23
24
20
21
22
23
24
25
21
22
23
24
25
26
12
13
14
15
16
17
13
14
15
16
17
18
14
15
16
17
18
19
15
16
17
18
19
20
16
17
24
25
17
18
25
26
18
19
26
27
19
20
27
28
42 non-ultimates
18 ultimates
7/3 ratio
3/2 ratio
3/2 ratio
28 non-ultimates
12 ultimates
7/3 ratio
42 non-ultimates
18 ultimates
7/3 ratio
3/2 ratio
3/2 ratio
28 non-ultimates
12 ultimates
7/3 ratio
Fig.11 Addition sub-matrices of the twenty fundamental numbers on 60 versus 40 numbers. Geometric variant of the remarkable identity(a + b)2 = a2 +
2ab + b2. Sous matrices d’addition des vingt nombres fondamentaux de 60 contre 40 nombres. Variante géométrique de l’identité remarquable
(a + b)2 = a2 + 2ab + b2 .
This configuration contrasts the upper 3/5ths of the six central columns of the matrix and the lower 3/5ths of the four
peripheral columns with the 2/5ths reciprocals of the columns considered. Also, in the right part of Figure 11, the vertical
mirror configuration to this arrangement generates the same oppositions in 3/2 ratios of the reciprocal non-ultimate numbers
and the reciprocal ultimate numbers of these other sub-matrices to 60 and 40 entities.
Also, by mixing, in Figure 12, the sub-matrices of 40 and 60 entities presented in Figure 11 and after having each split them
vertically into two equal parts to 30 and 20 entities, we obtain new matrices of 50 entities each. In these horizontal mirror
configurations, the non-ultimates and the ultimates are divided into exact ratios of value 1/1 with always 35 non-ultimates
versus 35 and always 15 ultimates versus 15. Also, by this geometric rearrangement, in addition to being configurations
horizontal mirror, the left configurations become vertical mirror of the right configurations and vice versa.
Mixed mirror configurations:
Mixed mirror configurations :
sub-matrix to 4
half mixed zones
(50 numbers)
← 1/1 ratio →
sub-matrix to 4
half mixed zones
(50 numbers)
sub-matrix to 4
half mixed zones
(50 numbers)
← 1/1 ratio →
sub-matrix to 4
half mixed zones
(50 numbers)
12
13
14
18
19
13
14
15
19
20
14
15
16
20
21
15
16
17
21
22
14
15
16
17
18
15
16
17
18
19
16
17
21
22
23
17
18
22
23
24
18
19
23
24
25
19
20
24
25
26
10
11
15
16
17
11
12
16
17
18
12
13
17
18
19
13
14
18
19
20
19
20
21
22
23
20
21
22
23
24
18
19
20
24
25
19
20
21
25
26
20
21
22
26
27
21
22
23
27
28
10
11
15
16
17
11
12
16
17
18
12
13
17
18
19
13
14
18
19
20
14
15
16
17
18
15
16
17
18
19
18
19
20
24
25
19
20
21
25
26
20
21
22
26
27
21
22
23
27
28
12
13
14
18
19
13
14
15
19
20
14
15
16
20
21
15
16
17
21
22
19
20
21
22
23
20
21
22
23
24
16
17
21
22
23
17
18
22
23
24
18
19
23
24
25
19
20
24
25
26
35 non-ultimates
15 ultimates
7/3 ratio
← 1/1 ratio →
← 1/1 ratio →
35 non-ultimates
15 ultimates
7/3 ratio
35 non-ultimates
15 ultimates
7/3 ratio
← 1/1 ratio →
← 1/1 ratio →
35 non-ultimates
15 ultimates
7/3 ratio
Fig.12 Mixed sub-matrices mirror of addition of the twenty fundamental numbers to 50 versus 50 numbers. Sous matrices mixtes miroir d’additions des
vingt nombres fondamentaux de 50 contre 50 nombres.
4.2 Concentric and eccentric matrices
In this matrix of the hundred additions of the twenty fundamentals, more sophisticated arrangements further oppose the
ultimate numbers and the non-ultimate numbers in exact 3/2 ratios. Thus, as described in the left part of Figure 13, five
concentric areas are opposed, three versus two, in the distribution of their ultimate numbers and their non-ultimate numbers in
3/2 ratios. The same phenomenon is reproduced by considering the five eccentric areas presented in the right part of Figure 13.
The eccentricity of these five areas is in total asymmetry with respect to the five initial concentric areas. However, as shown in
Figure 15, the same quantities of ultimates (and non-ultimates) are distributed in concentric or eccentric areas of the same size.
These five concentric and eccentric rings are by sizes whose respective values increase regularly according to this arithmetic
where x = 1: 4x 4(x+2) 4(x+4) 4(x+6) 4(x+8)
This arithmetic allows, in relation to the decimal system and by the interposition of the incorporated rings, the constitution of
sub-matrices, which oppose in size and in categories of numbers , in 3/2 value ratios.
Concentric configurations:
Eccentric configurations:
sub-matrix to 3
concentric zones
(60 numbers)
← 3/2 ratio →
sub-matrix to 2
concentric zones
(40 numbers)
sub-matrix to 3
eccentric zones
(60 numbers)
← 3/2 ratio →
sub-matrix to 2
eccentric zones
(40 numbers)
10
11
12
13
14
15
16
17
18
19
11
20
12
14
15
16
17
18
19
21
13
15
20
22
14
16
18
19
21
23
15
17
19
20
22
24
16
18
23
25
17
19
20
21
22
23
24
26
18
27
19
20
21
22
23
24
25
26
27
28
12
13
14
15
16
17
18
19
13
20
14
16
17
18
19
21
15
17
20
22
16
18
21
23
17
19
20
21
22
24
18
25
19
20
21
22
23
24
25
26
10
12
14
15
17
19
13
15
16
18
12
13
14
16
17
19
20
21
17
18
14
15
16
17
18
19
20
21
22
23
15
16
17
18
19
20
21
22
23
24
20
21
17
18
19
21
22
24
25
26
20
22
23
25
19
21
23
24
26
28
11
13
16
18
11
12
14
17
19
20
15
18
13
14
15
16
19
20
21
22
16
17
18
19
22
23
24
25
20
23
18
19
21
24
26
27
20
22
25
27
42 non-ultimates
18 ultimates
7/3 ratio
← 3/2 ratio
← 3/2 ratio →
28 non-ultimates
12 ultimates
7/3 ratio
42 non-ultimates
18 ultimates
7/3 ratio
← 3/2 ratio →
← 3/2 ratio →
28 non-ultimates
12 ultimates
7/3 ratio
Fig.13 From the addition matrix of the twenty fundamental numbers, concentric and eccentric configurations of sub-matrices to 60 and 40 entities
opposing their non-ultimates and their ultimates in 3/2 ratios. Depuis la matrice d’additions des vingt nombres fondamentaux, configurations
concentriques et excentriques de sous matrices de 60 et 40 entités opposant leurs non ultimes et leurs ultimes en ratios 3/2.
Also, by mixing, in Figure 14, the sub-matrices of 40 and 60 entities presented in Figure 13 and after having each split them
vertically into two equal parts to 30 and 20 entities, we obtain new matrices of 50 entities each. In these mixed configurations,
the non-ultimates and the ultimates are divided into exact ratios of value 1/1 with always 35 non-ultimates versus 35 and
always 15 ultimates versus 15. These reassemblings are exactly the same type as those proposed in Figure 12.
Mixed concentric configurations:
Mixed eccentric configurations:
5 half concentric mixed
zones (50 numbers)
← 1/1 ratio →
5 half concentric mixed
zones (50 numbers)
5 half eccentric mixed
zones (50 numbers)
← 1/1 ratio →
5 half eccentric mixed
zones (50 numbers)
10
11
12
13
14
11
16
17
18
19
12
14
15
16
20
13
15
18
19
21
14
16
18
20
22
15
17
19
21
23
16
18
21
22
24
17
19
20
21
25
18
23
24
25
26
19
20
21
22
23
15
16
17
18
19
12
13
14
15
20
13
17
18
19
21
14
16
17
20
22
15
17
19
21
23
16
18
20
22
24
17
19
20
23
25
18
22
23
24
26
19
20
21
22
27
24
25
26
27
28
10
12
14
16
18
13
15
17
19
20
12
13
14
16
18
17
19
20
21
22
14
15
16
17
18
15
16
17
18
19
20
22
23
24
25
17
18
19
21
23
20
22
24
26
27
19
21
23
25
27
11
13
15
17
19
11
12
14
16
18
15
17
19
20
21
13
14
15
16
18
19
20
21
22
23
20
21
22
23
24
16
17
18
19
21
20
22
24
25
26
18
19
21
23
25
20
22
24
26
28
35 non-ultimates
15 ultimates
ratio 7/3
← 1/1 ratio →
← 1/1 ratio →
35 non-ultimates
15 ultimates
ratio 7/3
35 non-ultimates
15 ultimates
ratio 7/3
← 1/1 ratio →
← 1/1 ratio →
35 non-ultimates
15 ultimates
ratio 7/3
Fig.14 From the addition matrix of the twenty fundamental numbers, concentric and eccentric configurations of sub-matrices to 50 entities each
opposing their non-ultimates and their ultimates in 1/1 ratios. Depuis la matrice d’additions des vingt nombres fondamentaux, configurations
concentriques et excentriques mixtes de sous matrices de chacune 50 entités opposant leurs non ultimes et leurs ultimes en ratios 3/2.
4.2.1 Arithmetic progressions
Other arrangements, as in the example in Figure 15, of three versus two concentric zones or three versus two eccentric zones
generate the same arithmetic phenomena with a 3/2 ratio between the respective quantities of ultimates of these zones sets.
This phenomenon is directly related to the regular progression of the value of quantities of ultimates from 2 to 10 depending on
the size of the concentric or eccentric zones considered.
Concentric configurations
Eccentric configurations
10
8
6
4
2
2
4
6
8
10
60 numbers
42 non-ultimates
18 ultimates
3/2 ratio
3/2 ratio
3/2 ratio
40 numbers
28 non-ultimates
12 ultimates
60 numbers
42 non-ultimates
18 ultimates
3/2 ratio
3/2 ratio
3/2 ratio
40 numbers
28 non-ultimates
12 ultimates
Fig.15 Regular arithmetic distribution of the ultimate numbers in the concentric and eccentric rings of the addition matrix of the twenty fundamental
numbers. Example of a 3/2 ratio arrangement (see Fig.13 also). Régulière répartition arithmétique des nombres ultimes dans les anneaux concentriques
et excentriques de la matrice d’additions des vingt nombres fondamentaux. Exemple d’arrangement de ratio 3/2 (voir aussi fig.13).
In each of the five concentric zones of the addition matrix of the twenty fundamental numbers, the quantity of ultimate
numbers (x) is linked to the whole quantity of numbers (z) of this zone by this formula:
x2 (x - 2)2 = z
In these same areas, the quantity of non-ultimate numbers (y) is linked to the quantity of ultimate numbers (x) by this formula:
x2 (x - 2)2 x = y
As demonstrated in Figure 16, this phenomenon remains identical for the five symmetrically eccentric zones.
quantity of numbers by zones* (z)
quantity of ultimate numbers (x)
quantity of non-ultimate numbers (y)
x2 (x - 2)2
= z
x
x2 (x - 2)2 x = y
22 (2 - 2)2
= 4
2
2
42 (4 - 2)2
= 12
4
8
62 (6 - 2)2
= 20
6
14
82 (8 - 2)2
= 28
8
20
102 (10 - 2)2
= 36
10
26
Fig.16 Arithmetic relationship between the value of the quantity of ultimates (and of non-ultimates) and the dimension
of the considered concentric * or eccentric * zone in the addition matrix of the twenty fundamentals. Relation
arithmétique entre la valeur de la quantité d’ultimes (et de non ultimes) et la dimension de la zone concentrique* ou
excentrique* considérée dans la matrice d’addition des vingt fondamentaux.
5. Matrix of the first hundred numbers
The study of the matrix of the first hundred whole numbers, configured in ten lines of ten classified numbers from 0 to 9,
reveals several singular phenomena according to the different classifications considered of the entities which compose it. These
phenomena will be introduced in different chapters including, to begin, this chapter distinguishing couples of ultimate numbers
from those without ultimates.
5. 1 Ultimate numbers and pairs of numbers
In the matrix of the first hundred numbers, 25 pairs of adjacent numbers, including at least one ultimate, are opposed, in an
exact ratio of 1/1, to 25 other pairs not including any. From the couple of numbers 0-1 to the couple of numbers 98-99, these
50 couples are always formed of two consecutive numbers as illustrated in Figure 17. Although 27 ultimate numbers are
present in the sequence of the first 100 numbers, only 25 couples integrating at least one ultimate emerge in this matrix. This is
due to the fact that the first four ultimate numbers are also the first four whole numbers and therefore that they are consecutive,
the first non-ultimate number (4) being in fifth position in the sequence of numbers. Also, only these last four are consecutive.
5. 1.1 Twice twenty-five pairs of numbers
As illustrated in Figure 17, it turns out that the 25 couples with ultimates and the 25 couples without ultimates oppose in 3/2
transcendent ratios according to whether they come from the upper part or from the lower part of the matrix of one hundred
first numbers. Thus, among the first 25 couples, in a ratio to 3/2, 15 consist of ultimates and 10 of non-ultimates and among the
last 25 couples, in a reverse ratio to 2/3, 10 consist of ultimates and 15 of non-ultimates.
15 couples
with ultimates
3/2 ratio
10 couples
without ultimates
0
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
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
25 couples
with ultimates
25 couples
without ultimates
↑ 3/2 ratio
↑ 2/3 ratio
10 couples
with ultimates
← 2/3 ratio
15 couples
without ultimates
Fig.17 Distribution of the 25 couples with ultimates and the 25 couples without ultimates in the matrix of the first hundred numbers.
Distribution des 25 couples avec ultimes et des 25 couples sans ultime dans la matrice des cent premiers nombres.
5. 1.2 Entanglement of pairs of numbers
Also, for each of the two upper and lower parts of this matrix, their splitting into two sets of 3 and 2 alternating lines as
illustrated in Figure 18 generates a multitude of entangled arithmetic phenomena always resulting in ratios of value 3/2 or
opposite ratios of value 2/3 simultaneously according to the zone considered and the nature of the couple considered (with or
without ultimates).
9 couples with ultimates
6 couples without ultimates
3/2 ratio
← 3/2 ratio →
← 3/2 ratio →
6 couples with ultimates
4 couples without ultimates
3/2 ratio
0
1
2
3
4
5
6
7
8
9
20
21
22
23
24
25
26
27
28
29
40
41
42
43
44
45
46
47
48
49
3/2 ratio
10
11
12
13
14
15
16
17
18
19
30
31
32
33
34
35
36
37
38
39
↑ 3/2 ratio *
2/3 ratio**
↑ 3/2 ratio*
2/3 ratio**
50
51
52
53
54
55
56
57
58
59
70
71
72
73
74
75
76
77
78
79
90
91
92
93
94
95
96
97
98
99
3/2 ratio
60
61
62
63
64
65
66
67
68
69
80
81
82
83
84
85
86
87
88
89
6 couples with ultimates
9 couples without ultimates
2/3 ratio
← 3/2 ratio →
← 3/2 ratio →
4 couples with ultimates
6 couples without ultimates
2/3 ratio
Fig.18 Strong entanglement of the distribution of 25 couples with ultimates and of 25 couples
without ultimate in the matrix of the first hundred numbers. Ratios between couples
respectively with ultimates* and without ultimate**. Forte intrication de la distribution des
25 couples avec ultimes et des 25 couples sans ultime dans la matrice des cent premiers
nombres. Ratios entre couples respectivement avec ultimes* et sans ultimes**.
5. 2 Sub-matrices of thirty versus twenty pairs of numbers
From the matrix of the first 50 pairs of whole numbers, in the sub-matrices made up of five vertically alternating zones to 3/5th
of column (30 pairs of numbers) such as those presented Figure 19 and in the complementary sub-matrices of five zones to
2/5th of column (20 couples), the quantities of couples with ultimates and those of couples without ultimate remain of equal
values and oppose in 3/2 ratios to the respective values of the complementary sub matrices.
Sub-matrix to 5 times
6 couples
30 couples (60 numbers*)
3/2 ratio
Sub-matrix to 5 times
4 couples
20 couples (40 numbers*)
Sub-matrix to 5 times
6 couples
30 couples (60 numbers*)
3/2 ratio
Sub-matrix to 5 times
4 couples
20 couples (40 numbers*)
0
1
4
5
8
9
10
11
14
15
18
19
20
21
24
25
28
29
30
31
34
35
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
62
63
66
67
72
73
76
77
82
83
86
87
92
93
96
97
2
3
6
7
12
13
16
17
22
23
26
27
32
33
36
37
60
61
64
65
68
69
70
71
74
75
78
79
80
81
84
85
88
89
90
91
94
95
98
99
2
3
6
7
12
13
16
17
22
23
26
27
32
33
36
37
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
64
65
68
69
70
71
74
75
78
79
80
81
84
85
88
89
90
91
94
95
98
99
0
1
4
5
8
9
10
11
14
15
18
19
20
21
24
25
28
29
30
31
34
35
38
39
62
63
66
67
72
73
76
77
82
83
86
87
92
93
96
97
15 pairs with ultimates
15 pairs without ultimates
1/1 ratio
← 3/2 ratio →
← 3/2 ratio →
10 pairs with ultimates
10 pairs without ultimates
1/1 ratio
15 pairs with ultimates
15 pairs without ultimates
1/1 ratio
← 3/2 ratio →
← 3/2 ratio →
10 pairs with ultimates
10 pairs without ultimates
1/1 ratio
Fig.19 Equal distribution of 25 couples with ultimates and of 25 couples without ultimate in sub-matrices to 30 versus 20 couples. Egale distribution
des 25 couples avec ultimes et des 25 couples sans ultime dans des sous matrices de 30 contre 20 couples.
Also, still from this same matrix of 50 couples of numbers, in symmetrical sub-matrices made up of 10 zones of 3 pairs of
numbers versus 10 zones of 2 pairs, as illustrated in Figure 20, the quantities of pairs of numbers with ultimates and those of
pairs without ultimate remain of equal values and oppose in 3/2 ratios to the respective values of the complementary sub-
matrices.
Sub-matrix to 10 times
3 couples (60 numbers*)
3/2 ratio
Sub-matrix to 10 times
2 couples (40 numbers*)
Sub-matrix to 10 times
3 couples (60 numbers*)
3/2 ratio
Sub-matrix to 10 times
2 couples (40 numbers*)
0
1
4
5
8
9
10
11
14
15
18
19
20
21
22
23
24
25
26
27
28
29
32
33
36
37
42
43
46
47
52
53
56
57
62
63
66
67
70
71
72
73
74
75
76
77
78
79
80
81
84
85
88
89
90
91
94
95
98
99
2
3
6
7
12
13
16
17
30
31
34
35
38
39
40
41
44
45
48
49
50
51
54
55
58
59
60
61
64
65
68
69
82
83
86
87
92
93
96
97
2
3
6
7
12
13
16
17
20
21
22
23
24
25
26
27
28
29
30
31
34
35
38
39
40
41
44
45
48
49
50
51
54
55
58
59
60
61
64
65
68
69
70
71
72
73
74
75
76
77
78
79
82
83
86
87
92
93
96
97
0
1
4
5
8
9
10
11
14
15
18
19
32
33
36
37
42
43
46
47
52
53
56
57
62
63
66
67
80
81
84
85
88
89
90
91
94
95
98
99
15 pairs with ultimates
15 pairs without ultimates
1/1 ratio
← 3/2 ratio →
← 3/2 ratio →
10 pairs with ultimates
10 pairs without ultimates
1/1 ratio
15 pairs with ultimates
15 pairs without ultimates
1/1 ratio
← 3/2 ratio →
← 3/2 ratio →
10 pairs with ultimates
10 pairs without ultimates
1/1 ratio
Fig.20 Equal distribution of 25 couples with ultimates and of 25 couples without ultimate in sub-matrices to 30 versus 20 couples. Egale distribution
des 25 couples avec ultimes et des 25 couples sans ultime dans des sous matrices symétriques de 30 contre 20 couples.
The existence of these singular arithmetic phenomena presented in this chapter greatly reinforce the main argument of this
study about whole numbers to merge the special numbers zero (0) and one (1) and the sequence of prime numbers into the
sequence of ultimate numbers. These phenomena indeed disappear completely without this fusion.
* Note: The configurations of the two types of sub-matrices presented in Figures 19 and 20 fits still in geometric variants of the
remarkable identity (a + b)2 = a2 + 2ab + b2 with a and b of respective value 6 and 4 (for entity quantities of 36 + 24 and 24 +
16).
6. The ten primordial ultimates
The identification of the first ten ultimate allows, with reference to the decimal system, to classify them as primordial. This
notion of primordiality is developed in Chapter 9.
6.1 Matrix of additions of the ten primordial ultimates
The cross additions of the first ten ultimates generate, Figure 21, a matrix of one hundred values including 30 ultimate numbers
(5x x = 6). Also these additions generate 30 digits.
30 digits including:
18 ultimates
3/2 ratio
12 non-ultimates
+
0
1
2
3
5
7
0
0
1
2
3
5
7
11
13
17
19
1
1
2
3
4
6
8
11
13
17
19
2
2
3
4
5
7
9
12