Tuesday, December 13, 2016

Prime Number Magic (3)

As well as getting separate estimates for the probability of occurrence of specific prime factors, one can also combine these numbers in every manner possible.

For example to find the probability that both a 2 and a 3 appear in the prime factor composition of a number, one needs to combine probabilities.

As we have seen - as regards - distinct prime factors, 2 appears on average in 1/2 of all numbers.

Again this is simply obtained as Po /(Po + 1) where Po represents the relative notion of the ordinal ranking of the prime in question.

And then based on complementary cardinal notions (of an absolute nature) we would expect 100 occurrences of 2 in every 100 numbers (i.e. 1/Pc – 1).

So 50 of these would relate to the number containing 2 as a (distinct) prime factor.

Based on cardinal notions 50 occurrences of 3 would on average be contained in a sequence of 100 numbers. And 2/3 of these would contain 3 as a distinct prime factor.

So 1/3 of all numbers on average would therefore contain 3 as a distinct prime factor.

So therefore the combined probability that a number would contain both 2 and 3 as distinct prime factors  = 1/2 * 1/3 = 1/6.

Therefore if one was to check through any orderly sequence of 100 natural numbers, one would find on average 1/6 = 17 (approx) to contain both 2 and 3 as prime factors.

Now if we were to be more restrictive and admit only those numbers that contain exactly one occurrence of both 2 and 3 as factors, we would have to then combine probabilities of obtaining exactly a single 2 and 3 in both cases.

So the probability of a number (already with 1 or more 2's) containing a single 2 as a prime factor = Po /(Po + 1)2 = 1/4 and the corresponding probability that of a number (already with 1 or more 3's) containing a single 3 (as a prime factor) = 4/9.

Then relating this to cardinal notion, in this case 1/4 and 2/9 of all numbers on average would contain exactly one 2 and exactly one 3 (as prime factors) respectively.

Therefore the probability of a number containing exactly one 2 and exactly one 3 (as factors) = 1/4 * 2/9 = 1/18.

Therefore in an orderly sequence of 100 numbers, one would expect on average 5 or 6 numbers to contain exactly a single 2 and a single 3 (as prime factors). 

One could also multiply each prime factor by its frequency of occurrence.

For example the frequency of occurrence (in relation to each number) with respect to 2 = 1.

So again we would expect 100 occurrences of 2 in every 100 numbers!

So 2 * frequency of occurrence = 2 * 1 = 2.

The corresponding frequency of occurrence with respect to 3 = 1/2

So 3 * frequency of occurrence = 3 * 1/2 = 1.5.

In general terms, the corresponding result for any prime factor = p *{(1/p – 1)} = p/(p – 1).

Therefore the value keeps falling as p increases with the result (in the limit) = 1.

Thus the greatest combined result (= 2) occurs with 2 with the value progressively falling for subsequent primes.

However a reverse picture unfolds when we confine ourselves to cases where just a single factor occurs exactly once.

So the frequency here for 2 = 1/4 (per number). And 2 * 1/4 = 1/2.

However with 3 the frequency (per number) = 2/9 and 3 * 2/9 = 2/3.

With 5 the frequency per number = 9/64  and 5 * 9/64 = .703...

So in this case the combined value keeps rising towards 1 as the value of p increases!

It is highly noteworthy that these results are based on the combination of cardinal with ordinal results.

In deeper philosophical terms, this implies the interaction of quantitative notions (of number independence) with complementary qualitative notions (of number interdependence) respectively.

From another perspective, we have the mixing of what - from a strictly analytic - perspective, appear as the paradoxical notions of randomness and order respectively.

So from the former perspective, the individual prime factors (comprising each natural number) appear as highly random; however from the complementary perspective, the sequence of natural numbers (derived from the collective relationship of these factors) appear as highly ordered.

However when we look at this relationship in true holistic terms i.e. from a two-way dynamic interactive perspective, randomness and order (like left and right turns at a crossroads) are seen as purely relative terms (that are mutually implied by each other).

Indeed this is true regarding the inherent nature of the primes and the natural numbers.

From the conventional analytic perspective - based exclusively on a restricted cardinal interpretation of number - the natural numbers are exclusively derived from the primes (as constituent factors).

However from the unrecognised  opposite perspective, natural number notions of order are implicitly contained in our very appreciation of the primes.

So for example when we speak of 2, 3, 5, 7 etc as the earliest examples of the primes, the natural number ordinal notions of 1st, 2nd, 3rd and 4th etc. are already contained - and vitally necessary - for cardinal appreciation.

Likewise from the opposite perspective, we look now each prime e.g. 3 as representing a group of members, the cardinal notion of 3 is necessarily already implicit in our appreciation of the ordinal relationship as between 1st, 2nd and 3rd members of this group.

Therefore though again - from a restricted analytic approach - the primes and natural numbers appear as separate entities with a one way relationship connecting them (through the unique product of prime factors) from a dynamic holistic approach, again the primes and natural numbers are seen clearly as but two sides of the same coin, which are mutually dependent on each other in both cardinal (quantitative) and ordinal (qualitative) terms.

We will look again at this more closely in the next blog entries.

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