Basic Prime Number 1

1.       A prime number, according to generally accepted definition, is divisible only by itself and by 1. Number 1 , however, complies with both parts of the definition, which therefore makes 1 a prime number by NATURE. But despite this, mathematic science today denies 1 its prime status, after consent had developed during the 20th century. But truth does not depend on human decisions: If 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". Obviously, for sheer undecisiveness, 1 must be carried along with the other two groups. However, this appears unsatisfactory.

3.       What seems necessary is a change of perspective: the priority of definition should not lie with the prime numbers, but the composite numbers which should be understood as particular positions of multiplicative number series, starting with number 1: Any individual number occurring is to be defined as single, for example, 5 is (1*)5. If it is doubled and trebled, it's 2*5 = 10 and 3*5 = 10. The first part of the multiplication series should be 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 is multiplier as well as multiplicand. In its first aspect 1 can be seen as a constant, and in its second aspect all successive numbers as multiplicands in unmultiplied singleness: 1*1, 1*2, 1*3 , 1*99 etc. As to number 1 the function of both multiplier and multiplicand potentially constitutes a SQUARE 1*1, and equal progression of multipliers and multiplicands creates more squares: 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 imaginal series of multiplication, for example 3*5 from preceding 2*5 or 3*4. Its result 15 can start another series of multiplication with multiplicands greater than 1.

A prime number does not contain two or more factors greater than 1. It is the beginning of a multiplication series (1)*PN, 2*PN etc.

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:

A composite number consists of two or more factors greater than 1.

A prime number consists of its own single factor.

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, which gives everything created its own image. The circle can be subdivided by three axes which can be completetd into a the hexagon and be extended to form 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:

prime number 1, nombre premier 1, numero primo 1, 1-13

The 13 numbers are placed subsequently from 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 and the 7 hexagonal points. It has to be examined what ORDER there exists between the two classes of numbers.

To begin with, the subsequent numeration in a symmetrical structure of points results in a ratio 6:7 correspondent to the numbers in both areas. So the two sums are 42+49. It turns out that the NS of 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 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.

Including the FV, the results are:

 

extension

hexagonal

 

 

ZZ

PZ

sm

ZZ

PZ

sm

GS

NS

21

21

42

28

21

49

91

FS

13

21

34

22

21

43

77

 

34

42

76

50

42

92

168

76:92 = 4*(19:23)

 

 

written: September 2018

Inhalt II