1

1

1

1

1

1

1

1

1

1

2

3

4

5

6

7

8

9

1

3

6

10

15

21

28

36

45

1

4

10

20

35

56

84

120

165

1

5

15

35

70

126

210

330

495

1

6

21

56

126

252

462

792

1287

1

7

28

84

210

462

924

1716

3003

1

8

36

120

330

792

1716

3432

6435

1

9

45

165

495

1287

3003

6435

12870

Now an especially interesting feature with respect to these
numbers can be made where n is a prime number. .
If we look at the entry with respect to n = 2, a unique
pattern unfolds, whereby numbers follow a cyclical pattern of 2 members, where
the 1^{st} when divided by 2 leaves a remainder of 1, and where 2 is
then a factor of the 2^{nd} number.
Then when we look at the entry with respect to n = 3,
another unique pattern unfolds, whereby numbers now follow a cyclical pattern
of 3 members, whereby the 1^{st} when divided by 3 leaves a remainder
of 1, with 3 then a factor of the other two numbers.
Finally to again illustrate, with respect to n = 5, another
unique pattern unfolds, whereby numbers follow a cyclical pattern of 5 members,
whereby the 1st when divided by 5 leaves a remainder of 1, with 5 a
factor of the other 4 members of the cyclical group.
Then when n is not prime, no such unique cyclical pattern
unfolds.
For example when n = 4, the 1st number (in the first group
of 4) leaves a remainder of 1 (when divided by 4). However the 1^{st}
number (in the next group of 4 i.e. 35) leaves a remainder of 3 (when divided
by 4).
Also, though 4 for example is a factor of the second member
(of the first group of 4) it is not a factor of the 3^{rd} member.
So again in an “Alice in Wonderland” type manner this
provides in principle a valid manner for testing whether any number is prime,
which in general terms can be expressed as follows.
With respect to the unique number sequences associated with
the polynomial equation,
(x – 1)^{n }=
0, for n = 1, 2, 3, 4,…,
if n is prime then a unique cyclical sequence of n
individual members will unfold, whereby the 1st member of each cyclical group
will always leave a remainder of 1 (when divided by n) with n then being a
factor of the other (n – 1) members of the group.
And if this is not the case, then n is not a prime number.
As the unique number sequences involved are potentially
infinite, this would then strictly prove a neverending process.
However if the required pattern for prime numbers applies to
the 1st cyclical group of members involved, then it will apply to all further groups.
Therefore we can more simply restate the position as
follows.
With respect to the unique number sequences associated with
the polynomial equation,
(x – 1)^{n }=
0, for n = 1, 2, 3, 4,…,
if n is prime, then a unique cyclical sequence of n
individual members will unfold, whereby the 1st member of the first cyclical
group will leave a remainder of 1 (when divided by n) with n then a
factor of the other (n – 1) members of the group.
For example if 7 is prime, the unique cyclical sequence of
7 individual members will unfold, whereby the 1st member (i.e. 1) of the first
cyclical group leaves a remainder of 1 (when divided by 7) with 7 then being a
factor of the other 6 members of the group.
And as we can see from the table, these six members are 7, 28,
84, 210, 464 and 924 respectively with 7 a constituent factor of all these
numbers.
So 7 therefore is a prime number. (Incidentally the next
number, which is then the 1^{st} member of the next cyclical group i.e.
1716 as expected, then leaves a remainder of 1, when divided by 7).
However 8 clearly is not prime. Whereas the 1^{st} member
 as is always the case  leaves a remainder of 1 (when divided by 8), 8 is not
a factor for example of the second i.e. 36 of the remaining seven members of
the group.
As I have mentioned on many occasions in these blog entries,
there are really two complementary ways of defining a prime number.
The standard (Type 1) approach, based on cardinal number identity,
defines a prime as an indivisible “building block” of the natural number system,
so that it has no constituent factors (other than itself and 1). This
corresponds directly with the quantitative aspect of number.
However the alternative (Type 2) approach based on ordinal
number identity defines a prime in reverse as a group that is already composed of natural
number members.
So for example 5 as a prime number is thereby necessarily
composed of its 1^{st}, 2^{nd}, 3^{rd}, 4^{th},
and 5^{th} members.
This directly corresponds with the qualitative aspect of
number, which then indirectly can be expressed in quantitative fashion through
obtaining the 5 roots of 1.
Now interestingly as one of the n roots of 1 is always 1,
this thereby is not unique.
So for a prime number n, the remaining (n – 1) roots are
unique.
So what we have uncovered therefore through these unique
number sequences of a simple type of polynomial equation, is an
approach to the primes that reflects the ordinal (qualitative) rather than
cardinal (quantitative) definition of a prime.
So once again from the standard cardinal perspective, a
number n is prime if it has no constituent prime factors.
However, from this alternative ordinal perspective, a number
n is prime if it is a constituent factor of a unique group of n – 1 members (with
the remaining member = 1).
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