Remarkable Features of the Number System (4)

I next studied the factor composition for those natural numbers with repeating prime structures.

For example from my prime factor generator at the 15 digit level,

900000000000414 = 22441 * 7549 * 4157 * 71 * 3 * 3 * 2.

So this represents a natural number with repeating prime structures that contains in all 7 prime factors. (1)

However if we were to count factors (where each distinct prime is counted only once) we would then have 6 prime factors ( as 3 recurs twice). (2)

Finally if we were to count factors where any recurring factor is excluded from consideration, we would obtain just 5 prime factors. (3)

So I sought to find with respect to the natural number system as a whole, the relationship between (1) and (2), then the relationship between (2) and (3) and finally the relationship as between (1) and (3).

Though still at a comparatively early stage of the number system - where numbers typically are still composed of relatively few prime factors - I could see a distinctive pattern beginning to emerge, which I will now outline for further empirical and theoretical investigation.

Therefore with respect to the combined factors of all natural numbers with repeating prime structures, the ratio of (1) where all the prime factors are included to (2) where any recurring prime is counted just once, ~ √2 (when n is sufficiently large).

The corresponding ratio of (2) where again each recurring prime is counted just once to (3). where recurring primes are excluded entirely from consideration, also  ~ √2 (when n is sufficiently large).


This therefore directly implies that the corresponding ratio of (1), where once more all prime factors are included to (3) where recurring prime factors are completely excluded   ~  2 (when n is sufficiently large).

Another way of expressing this result is that with respect to the average number of prime factors (for each natural number with a repeating prime structure) that the ratio of (1) to (2) ~ √2, that the ratio of (2) to (3) also ~ √2 and that finally the ratio of (1) to (3) ~  2 (when n is sufficiently large).

Expressed in yet another alternative manner, this would imply again - with respect to the natural numbers with repeating prime structures - that the average number of non-repeating prime factors would approximately equal the corresponding average number of recurring prime factors. And this approximation would become ever more accurate as n increases.

And because the overall number of factors for natural numbers with repeating and non-repeating prime structures respectively approaches equality (as n increases), this would therefore imply that the non-recurring prime factors (with respect to the natural numbers with repeating prime structures) would approximate 1/2 the total for corresponding factors of natural numbers (with non-repeating prime structures).

Remarkable Features of the Number System (3)

As is well known (Hardy-Ramanujan theorem), the average number of (distinct) primes ~ log log n (for n sufficiently large).

Therefore, the corresponding combined frequency of (distinct) prime factors up to n ~ n(log log n).

However, this in turn can be broken down into (distinct) factors associated with natural numbers which have both non-repeating and repeating prime structures respectively.

So, based on findings already reached, the combined frequency to n, of factors for the (natural) numbers with non-repeating prime structures
~ n(log log n)/(1 + 2/π).

The corresponding combined frequency for the (natural) numbers with repeating prime structures

~ n(log log n)/(1 + π/2).

Therefore, the total combined frequency for (natural) numbers with both non-repeating and repeating prime structures respectively ~ n(log log n)/(1 + 2/π) +  n(log log n)/(1 + π/2), i.e.  n(log log n)!


In the remainder of this blog entry, I will attempt to probe the deeper holistic mathematical explanation as to why π is so intimately involved in a simple manner in the results established.

Once again the conventional quantitative approach to the primes is of a strictly limited linear nature, that fails to recognise the inherent dynamic interactive nature of the relationship of the primes with the natural numbers.

Put another way - using the terminology frequently adopted on these blog entries - it is strictly of a Type 1 nature (based on quantitative notions of number independence).

However there is a corresponding (neglected) complementary aspect for every number relationship, which is of a Type 2 nature (based on qualitative notions of number interdependence).

Therefore, in truth, all number interactions are inherently dynamic combining the complementary interaction of both the Type 1 and Type 2 aspects.


Now, again briefly in Type 1 terms, 30 is represented as the combination of distinct prime factors  as

(2 * 3 * 5)¹  = 30¹.

However in Type 2 terms, 30 is represented in an inverse manner as

1^{(2 * 3 * 5)}  = 1³⁰.

Whereas the first (Type 1) expression is directly understand in the conventional linear rational manner, the latter (Type 2) expression - relating directly to intuitive holistic appreciation - is indirectly expressed in a circular paradoxical rational manner as the 30 roots of the unit circle (in the complex plane).

These roots (as equidistant points on the unit circle) then provide the appropriate means of indirectly expressing ordinal notions of number interdependence. 

In other words, these 30 roots express the notions of 1st, 2nd, 3rd,....,30th (in the context of 30 possible positions).

So therefore, quite simply, both cardinal and ordinal notions of number, which mutually imply each other in a dynamic complementary manner, entail both the linear and circular notions of number (in Type 1 and Type 2 terms).

And corresponding to linear and circular notions in a quantitative  terms are complementary linear  and circular notions in a qualitative manner.

In other words, the proper dynamic understanding of number entails both linear (analytic) and circular (holistic) modes of interpretation respectively.

Now, the very nature of π, in quantitative terms, expresses the pure relationship as between the circumference of the circle and its line diameter.

In corresponding qualitative  terms, π expresses the pure relationship as between circular (holistic) and linear (analytic) type understanding of the number system.

In other words, in dynamic interactive terms, a perfect synchronicity characterises the relationship as between the primes and natural numbers - throughout the number system - in both quantitative (cardinal) and qualitative (ordinal) terms (which are dynamically interdependent).

Thus the simple expressions involving π, that I have used here to describe the relationship of non-repeating to repeating primes (in its various manifestations) perfectly illustrates the dynamic synchronistic behaviour that ultimately characterises the true nature of the number system.

And this will never be properly appreciated from the conventional mathematical perspective! 

This is why I stress, again and again, that nothing less than a total revolution in the present accepted manner of understanding Mathematics is now urgently required.

Remarkable Features of the Number System (2)

We have seen in the last entry that the average gap as between natural numbers with non-repeating prime structures ~ 1 + 2/π.

The corresponding average gap as between natural numbers with repeating prime structures
~ 1 + π/2.

This therefore entails that the frequency of the natural numbers up to n with non-repeating prime structures ~ n/(1 + 2/π).

The corresponding frequency of the natural numbers up to n with repeating prime structures ~ n/(1 + π/2).

I next looked at the combined number of factors for numbers with repeating structures as opposed to those with non-repeating prime structures.

In the earlier stages of the number system, the combined frequency of factors for natural numbers with repeating prime structures predominates over those with non-repeating structures. However as one ascends the number scale, this imbalance starts to steadily fall. So whereas initially - say from 1000 to 2000 - the ratio of combined factors of numbers with non-repeating to numbers with repeating prime structures is about .75, higher up the number system (in the vicinity of 10¹⁵) the corresponding ratio is close to .9.

Now this might seem like a slow increase relative to the size of natural numbers involved. However relative to the combined number of factors it is in fact very rapid. For it must be remembered that the average number of distinct prime factors per (natural) number rises very slowly (i.e. at the rate of log log n)!

So the assumption here is that with a sufficient increase in the average frequency of prime factors (for each number) that the ratio of the combined frequency of factors - for natural numbers with non-repeating prime structures - to that with repeating structures approaches 1.

This would then further imply that the  ratio of the average number of factors - for each individual natural number with repeating prime structures - to that with non-repeating structures ~ 1 + 2/π.

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!