A Special Case (4)

I am going back again to that grid table that lists the first 9 numbers in the unique sequences associated with the equation (x – 1)ⁿ = 0 from n = 1 to 9.

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 never-ending 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).