Wednesday, February 10, 2016

Remarkable Features of the Number System (1)

As is well known, every natural number (except 1) represents a unique combination of prime factors.

However, we can make a distinction here as between the natural numbers that are based on repeating and non-repeating structures respectively.

A repeating structure entails at least one prime that occurs more than once!

So for example 28 = 2 * 2 * 7 represents a repeating prime structure; however 30 = 2 * 3 * 5 represents a non-repeating prime structure.

In this context, I thought that it might be interesting to study the overall behaviour withing the (natural) system of numbers based on these two prime structures.

In this regard, I was greatly assisted by a prime factor generator that could provide the factors of all natural numbers up to 1015.

I initially studied carefully the first 1000 numbers and found to my great surprise that a remarkable consistency with respect to distribution prevailed.

For convenience, I broke this interval of 1000 into ten intervals of 100. then with respect to numbers comprising non-repeating prime structures I found that the average seemed to be settling down very close to 61 per 100. Indeed little variation was in evidence with the smallest sample producing 59 such numbers and the the largest 64!.

However this still represented a very early stage of the number system (with a small number of prime factors involved).

I thought therefore that this pattern would change considerably higher up the number system. Intuitively it then seemed to me that with larger numbers, repeating prime structures would begin to predominate. Therefore, on this basis one would expect the frequency of numbers with non-repeating primes to steadily fall!

However, to my considerable surprise, this was not the pattern that unfolded.

Thus when I studied another 1000 numbers from 150,000,000,000,001 - 150,000,000,001,000 the pattern basically remained unchanged. So again, the average per 100 of numbers with non-repeating primes stayed very close to 60 (with all samples falling between 58 and 64).

I also made numerous other investigations at varied intervals of this number range i.e. 1 - 1015 to find the same pattern occurring with remarkable stability.

This therefore strongly suggested to me that the ratio of numbers (with non-repeating prime structures) with respect to the overall number system is governed by a constant.

So the next question was which constant fitted the bill!

Happily, a simple intuition here (based on the importance of π with respect to prime number distributions), guided me directly to what I believe now is the correct position.

I will state this position now!

The ratio of the natural numbers to those with non-repeating prime structures = 1 + 2/π (i.e. 1.6366... approx).

The corresponding ratio of the natural numbers to those with repeating prime structures = 1 + π/2 (i.e. 2.5707... approx).

Then, the ratio internally of those natural numbers with non-repeating prime structures to those with repeating prime structures = π/2.

Another way of expressing the first two results is to say that the average gap as between natural numbers (comprised of non-repeating prime structures) = 1 + 2/π.

And the average gap as between natural numbers (comprised of repeating prime structures) = 1 + π/2.

Now clearly over any finite stretch of the number system, these results can only be approximated rather than exactly attained, though of course we would expect the approximations to continually improve with more data.

In one way, I find these important regularities - representing the fundamental behaviour of prime factor combinations - to be stunning in their simplicity and am amazed that I have never seen them highlighted before!

Now of course - much like the early history of the prime frequency approximation to n i.e. n/log n - this finding is based on empirical testing rather than definitive proof. However that does not lessen its potential significance for the nature of the number system!

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