Prime Number 1

1.       A prime number, according to worldwide consensus, is a number greater than 1 that is divisible only by itself and by 1. That excludes 1 as a prime number. But truth does not depend on human decisions: Whether 1 IS prime or not, is a logical and an ontological problem. In fact, according to Wikipedia-information 1 could be found in lists of prime numbers until 1956. It is to be considered whether the current definition of prime numbers has the right logic on its side.

2.       The reason generally forwarded for the exclusion of number 1 from the class of prime numbers is that the factors of composite numbers would not be written unequivocally as 2*3 for example, but also as 1(n)*2*3.

So what happens is that in order to make the definition of prime numbers as simple as possible, number 1 must quit the field. In fact, according to the prevailing definitions, there aren't just two, but three groups of numbers: "composite, prime, and the unit 1". This makes things complicated again.

3.       What seems necessary is a CHANGE OF PERSPECTIVE: the priority of definition should not lie with the prime numbers, but the composite numbers. A composite number is to be understood as a particular position in a multiplication series, starting with number 1. Each number N on the one hand stands for itself with its own identity and meaning, on the other hand it forms the beginning of a potential series of multiplication, which is necessarily represented as  1*N. If for example 5 is doubled and trebled, it's 2*5 = 10 and 3*5 = 15. The first factor of the multiplication series is called MULTIPLIER, the second MULTIPLICAND. So 10 would be the second successive result in multiplying 5. The multiplicator 1 is IMMANENT to the initial multiplicand number and so NEED NOT be placed in front of it.

The basic number 1 itself can be multiplier as well as multiplicand. As a multiplier it occurs only once, and a multiplicand it is a constant in successive groups of ones: 1*1, 2*1, 3*1. Equal multipliers and multiplicands produce SQUARES, e.g. 1*1, 2*2, 3*3 etc.

4.       Composite numbers and prime numbers consequently should be defined as follows:

A composite number consists of two or more factors greater than 1 within an imaginary series of multiplication, for example 3*5 follows preceding 2*5 or 3*4. Its result 15 can start a new series of multiplication starting with 1*15.

As the current definition of prime numbers is inadequate, it has to be reformulated. The order should be composite numbers first and prime numbers second:

The new definition eliminates the aspect of division as unessential.

5.       For mathematicians numbers are a matter of axiomatic theory, i.e., they do not concern themselves with the origins of numbers and their possible meanings and structures. So it eludes their attention that the decimal system contains innumerable well-ordered proportions and relationships. What could be the sense of two classes of numbers if there didn't exist a wise system of order. One example is the composite numbers (CN) and prime numbers (PN) of 1-13 in subsequent additions:

CN

 

 

 

4

 

6

 

8

9

10

 

12

 

49

PN

1

2

3

 

5

 

7

 

 

 

11

 

13

42

49:42 = 7*(7:6)

6:7 numbers bring about an inversive relation of 7:6. Without 1 no relation would be possible.

Furthermore, every NUMERIC VALUE (NV) is matched by a FACTORAL VALUE (FV). The relationship of the two values can be written as an equation: 6 = 2*3: NV = 6, FV = 2+3 = 5; 5 = (1)*5: NV = 5, FV = 5; 1 = (1)*1: NV = 1, FV = 1. Numerical values and fact oral values of prime numbers are identic. As to the numbers 1-13 the following relations emerge:

 

CN

sm

PN

sm

GS

NV

4

6

8

9

10

12

49

1

2

3

5

7

11

13

42

91

FV

4

5

6

6

7

7

35

1

2

3

5

7

11

13

42

77

 

48

36

84

36

48

84

168

35:42 = 7*(5:6); 77:91 = 7*(11:13)

The numeric sums (NS) + factoral sums (FS) of the 6 CN and 7 PN are equally 84. The NS+FS of the first 9 (4-9, 1-7) and the last 4 (10-13) numbers are 84 (48+36; 36+48) again.

The total of numbers within the decimal system proves to be a network of innumerable relations, which would seriously be impaired without 1 understood as prime number.

6.       On an ontological level the FIRST of all numbers is the origin of everything being. Among geometrical figures the CIRCLE symbolises eternity most clearly. It can be subdivided by three axes and completed into a hexagon and furthermore EXTENDED into a hexagram containing two TETRACTYS. A tetractys is an equilateral triangle consisting of 1+2+3+4 points, well-known from the Greek mathematician Pythagoras. The entire figure of hexagram consists of 13 points:

The 13 numbers are placed from left to right and top to bottom. The inner and outer circle are in ratio 1:3. There are composite numbers (blue) and prime numbers (orange) both on the 6 extension points and the 7 hexagonal points. It has to be examined what ORDER there exists between the two classes of numbers.

To begin with, the regular numeration results in a regular ratio 6:7 of the two sums 42 and 49 correspondent to the 6 numbers in the extension area and 7 in the hexagonal areas. The composite and prime numbers in both areas are divisible by 7:

 

extension

sm

hexagonal

sm

tot.

cn

9

12

 

 

21

4

6

8

10

28

49

pn

1

2

5

13

21

3

7

11

 

21

42

 

 

 

 

 

42

 

 

 

 

49

91

The table is arranged according to composite numbers (cn) and prime numbers (pn). The two sums form the ratio 42:49, which can be written as 7*(6:7).

The sum 42 of the 6 extension numbers is the same as the 7 prime numbers between 1 and 13, and vice versa the 7 hexagonal numbers agree with the sum 49 of the 6 composite numbers.

 

written: September 2018

Inhalt II