Do Numbers Evolve? (4)

I have referred repeatedly to the dynamic interaction as between quantitative and qualitative aspects with respect to number.

Ultimately this interaction relates to the interplay of both the finite (actual) and infinite (potential) notions, which in psychological terms relate to both conscious and unconscious aspects of understanding respectively.

So mathematical objects such as numbers possess an actual existence from a finite (conscious) perspective directly mediated in rational terms; however equally they possess a potential existence from an infinite (unconscious) perspective that is directly mediated in an intuitive manner. And both of these ceaselessly interact dynamically in experience leading to continual transformation with respect to such objects.

So properly, i.e. in a dynamic interactive manner, number thereby necessarily evolves. And this relates not just to the nature of (internal) psychological understanding, but also to the external objects (both of which - by definition - are now necessarily relative to each other).

However as an alternative to the sole use of  quantitative and qualitative terms, I would suggest the corresponding pairing of analytic and holistic (which perhaps appears a little more scientific).

However it is important to point out that I am using analytic in the broader sense in which the terms is commonly used in science, which equates directly with a reduced quantitative interpretation of relationships!
Now analytic has also a well-defined narrower meaning within Mathematics in relation to the treatment of infinite series and limits. However suffice it to say that within Mathematics, more restricted use of the terms "analytic" (and "analysis") are also analytic in the broader sense of the term (in that they are defined solely within a reduced quantitative context).

Therefore to return to my basic position, properly understood all number has both analytic and holistic aspects (in dynamic relationship with each other).

From one important perspective, this is true internally for each number. So, as we have seen the number "2" for example entails both the analytic aspect of "2" as a specific number quantity in cardinal terms, and the holistic aspect of "2" (i.e. twoness) as collectively applying to all possible instances of "2").
So properly understood these two notions are actual and potential with respect to each other.

And because in the dynamics of experience (like approaching a crossroads from opposite directions) polar reference frames continually switch) there is also an important sense, where "2" now refers to a specific number quality (i.e. in the ordinal notion of 2nd) while "2" now attains a collective meaning in the cardinal notion of dimension that now actually applies to all numbers.

Thus in the dynamics of the experience of each number, there is a ceaseless two-way interplay of both analytic and holistic type understanding, through which we are enabled to switch seamlessly as between cardinal and ordinal type appreciation (with respect to both objects and dimensions).


Then from the other important perspective, similar dynamics apply to the number system as a whole.
This then enables us to consistently combine both the cardinal and ordinal identities of all numbers (not is relative isolation) but in full relationship with other numbers.

Now the precondition for such consistency is that a seamless means exists for switching as between both the Type 1 and Type 2 aspects of the number system.

Thus from one perspective we need to be able to seamlessly convert the Type 2 aspect in a Type 1 manner.
Then equally from the alternative perspective we need to be able to seamlessly convert the Type 1 aspect in a Type 2 manner.


Though its significance seems to me to be completely missed by the mathematical community, I will start with the first of these conversions (which in fact is relatively easy to appreciate).

Now we will illustrate here again for convenience with respect to the number "2".

So the standard analytic definition of "2" (as a specific number quantity) is given through the Type 1 aspect as 2¹. So once again the Type 1 aspect is always defined with respect to the default dimensional value of 1.

The corresponding holistic definition of "2" (as the collective number quality of twoness) is given through the Type 2 aspect as 1². "2" now refers directly to a number dimension (rather than a base quantity).

Thus to convert this Type 2 aspect in Type 2 terms, we need in effect to obtain the square root.

So in general terms xⁿ = 1 with in this case x² = 1. So x = + 1 and – 1.

We have now moved to a circular definition of number (with both + 1 and – 1 lying on the unit circle in the complex plane).

However these two results are given but an analytic quantitative interpretation in conventional mathematical terms.
However the corresponding holistic meaning is highly revealing, requiring in effect a uniquely distinctive manner of mathematical interpretation.

+ in this context entails the psychological notion of positing (i.e. making conscious).
– however entails the corresponding notion of negation (i.e. of what is unconscious) thereby representing unconscious understanding.

When understanding is especially refined, as with the fusion of matter and anti-matter particles in physics, unconscious negation (of what is consciously posited) will approach full attainment resulting in a pure intuitive understanding (representing a psycho spiritual energy state).

So strictly speaking the holistic appreciation of each number represents a pure energy state (with complementary physical and psychological meanings).

Thus in effect we have two extremes with respect to the understanding of number (and remember in dynamic terms number as object has no strict meaning independent of such understanding)!

Thus we can appreciate number in the standard analytic fashion as an absolutely existing quantity form (that never changes). Here it is viewed as nothing in qualitative terms

However from the opposite extreme we can appreciate number in the unrecognised holistic fashion as approaching a pure energy state (where it is nothing in quantitative terms).

However properly understood, number experience entails an interaction somewhere between both extremes, where both quantitative aspects (as form) and qualitative aspects (as energy) ceaselessly interact leading to a continual transformation thereby in the nature of each number.

So once again we have the analytic quantitative extreme (recognised through the Type 1 aspect)

Here 2 = 1 + 1 (Strictly 2¹ =  1¹ + 1¹⁾.

So here the two units are defined in a homogeneous quantitative manner (i.e. without any distinctive quality)

Then in the Type 2 system 2 = 1st + 2nd (so both units are now defined as without any quantitative distinction!)

Then when we convert the Type 2 to the Type 1, we can indirectly represent this important reality consistently in a quantitative manner.

So 1st and 2nd are now represented as + 1 and .– 1 respectively.

And + 1 .– 1 = 0!

So the task of converting consistently from Type 2 to Type 1 implies that we can represent the ordinal members of each group uniquely by a set of circular numbers (lying as roots on the unit circle) that always add up to zero.


And this is where the prime numbers can be seen to have an equally valid Type 2 (as well as Type 1) identity.

From the Type 1 perspective, the unique importance of the primes comes from viewing them as the "building blocks" of the natural number system.

So all natural numbers (other than 1) can be uniquely expressed as the product of prime factors.  

However, the primes have an equally important role in Type 2 terms, where however their directional link to the natural numbers is completely reversed.

So from the Type 2 perspective, each prime can be uniquely expressed in an ordinal natural number fashion by its various roots (again except 1).

So for example if we take 5 as a prime, it can be uniquely expressed in terms of its 5 roots (excluding 1 which is common to all roots).

Now these 5 roots provide an indirect (Type 1) means of uniquely expressing in quantitative terms   the various natural number members of 5 (i.e. 1st , 2nd , 3rd, 4th and 5th respectively) in an ordinal manner.

However there is an obvious paradox with respect to the Type 1 and Type 2 approaches.

In the first case, each natural number (except 1) is uniquely defined by its prime members in cardinal terms.

In the 2nd case, each prime is uniquely defined by its natural number members (except 1) in ordinal terms (indirectly expressed in a quantitative manner through its prime roots).

This leads directly to the holistic qualitative recognition of the two-way interdependence of primes and natural numbers in both cardinal and ordinal terms.

In other words a holistic synchronicity entailing the two-way interaction of primes and natural numbers (which is directly qualitative in nature) underlies the deepest workings of the number system.

However though obvious when viewed from the appropriate perspective, the realisation of  this simple fact will permanently elude a mathematical profession that reduces interpretation of number in a merely quantitative fashion.

Do Numbers Evolve? (3)

Just to recap briefly from yesterday's entry!

Every number has two distinctive meanings. So 2, for example represents a specific quantity in cardinal terms; however equally it represents a collective dimensional quality as "twoness" (that potentially applies to all specific quantities).

And both of these meanings in experiential terms are dynamically inseparable from each other.
So every number therefore represents a dynamic interaction with respect to both its quantitative and qualitative aspects (which are complementary).

Then in the dynamics of experience, reference frames continually switch. So 2 now attains a specific quality as 2nd (i.e. the ordinal nature of 2) while the dimensional notion of 2, in complementary fashion, assumes a cardinal identity (which is the conventional meaning associated with a number representing a power or exponent).

So rather than just one unambiguous natural number system that  can be unambiguously defined in rigid absolute terms as;

1, 2, 3, 4,.....,

we now have two complementary aspects of the number system which dynamically interact with each other.

Thus to identify number with its specific quantitative aspect, we assume a default fixed dimensional value of 1.

Therefore, from this perspective, the numerical value of an expression entailing higher powers (i.e. dimensions) is thereby reduced in a 1-dimensional manner.


This quantitative aspect is then represented as:

1¹, 2¹, 3¹, 4¹,.....,


Then in reverse manner to identify number with its collective qualitative aspect, we now in complementary fashion, maintain the base number fixed at 1, while allowing the dimensional value to vary through the natural numbers.

So from this alternative qualitative perspective, the number system is defined as:

1¹, 1², 1³, 1⁴,.....,

I refer to these two aspects as Type 1 and Type 2 respectively.

Both can only be properly understood (thereby mirroring authentic experience) as in dynamic complementary relationship with each other i.e. as quantitative to qualitative (and qualitative as to quantitative respectively).


The quantitative (cardinal) aspect is defined strictly without qualitative meaning.

Thus from this perspective 2 = 1 + 1 (i.e. 2¹ = 1¹ + 1¹).

Thus the two units here are fully homogeneous in quantitative terms (thereby lacking qualitative distinction).

It is the reverse from the opposite ordinal perspective.

Here the two units are represented in qualitative terms as 1st and 2nd (thereby lacking any quantitative distinction).

When one clearly realises that in truth all number operations properly entail both quantitative and qualitative aspects in dynamic relationship with each other, then the key issue arises as to consistency as between both sets of meanings.

This entails that a satisfactory way of converting from quantitative to qualitative (and qualitative to quantitative respectively) necessarily must exist if we are to maintain true confidence in all subsequent operations.

And this is what the zeta zeros essentially relate to, though this is not all yet realised due to the strongly reduced (i.e. merely quantitative) nature of accepted mathematical interpretation.


However just as we have two aspects to the number system, likewise properly we should have two sets of zeta zeros.

I refer to these two two sets as Zeta 1 and Zeta 2 respectively

Now the first set (i.e. Zeta 1) can be identified directly with the Riemann zeros.

However there is an alternative - and simpler - set, whose true function is not properly realised. (which I refer to as Zeta 2).

When we refer back once again to yesterday's blog, I identified two key quantitative/qualitative type relationships with respect to the number system.


So again, illustrating with reference to the number "2", I stated that two notions, which are quantitative and qualitative with respect to each other, are necessarily involved.

Thus we have the specific quantitative notion of 2 in relation to a general collective qualitative notion (as "twoness").

Then when we switch the frame of reference (as in the manner of approaching a crossroads from the opposite direction) we  now obtain a specific qualitative notion of 2 (as the ordinal notion of 2nd) with respect to a collective quantitative notion of 2 (as representing power or dimension).

Note once again how these dynamics are completely short-circuited from the conventional perspective with the quantitative notion of number (both as base number and dimension remaining). So ordinal notions in conventional mathematics are treated in a merely reduced fashion as "rankings" based on cardinal understanding  of  a quantitative nature!


Now in basic terms, as we shall see, the Zeta 2 zeros refer directly to conversion as between quantitative and qualitative interpretation with respect to individual numbers (such as "2").

However we also saw that a more general problem exists with respect to the collective relationship of numbers to the number system (in quantitative and qualitative terms).

Therefore the quantitative general notion of "a number" has strictly no meaning in the absence of the corresponding qualitative notion of "numberness" (that potentially applies in all cases).

Now the famed Riemann zeros (which I refer to as the Zeta 1 zeros) properly relate to this more general problem with respect to the number system as a whole, of ensuring consistency with respect to both the quantitative and qualitative interpretation of number. In direct terms they provide a means of converting as between quantitative and qualitative type usage.

Put another way - both with respect to any specific number and numbers generally - the notion of quantitative independence has no meaning in the absence of the corresponding notion of qualitative interdependence  (that thereby enables numbers to be related to each other).

Therefore we can only properly understand the number system in a dynamic relative manner (entailing the complementary notions of independence and interdependence respectively).

Once again even momentary reflection on the matter should immediately suggest to one that there is something fundamentally wrong with conventional mathematical interpretation.

We insist on interpreting numbers in an absolute independent manner (i.e. with respect to their mere quantitative characteristics). However this begs the obvious question of how numbers can then be related with each other (which assumes some quality of interdependence).

However because such reduced interpretation has now become so ingrained due to an unquestioned consensus, the mathematical community remains blind as to this must fundamental of all issues!

Do Numbers Evolve? (2)

In the last blog entry I argued that the conventional belief in the absolute existence of number is untenable from an experiential perspective.

So all numbers possess both external (objective) and internal (mental) aspects which dynamically interact.

Thus the conventional view of number represents but a special limiting case where both poles are fully abstracted from each other. Now this cannot of course completely occur in experiential terms (which would render understanding of number impossible); however it can be approached in a relative manner.

Thus the conventional absolute view of number (as rigid unchanging entities) is then appropriately understood as just one special - though admittedly important - limiting case with respect to interpretation.

As I have frequently stated this is directly associated with linear (1-dimensional) understanding based on interpretation within single isolated polar reference frames.

So for example the conventional treatment of number in merely quantitative terms - rather than a relationship entailing both quantitative and qualitative aspects - represents such linear interpretation.

However when we recognise the truly relative nature of mathematical understanding, as the interaction of opposite poles such as external and internal, this opens up entirely new vistas where the number can be given a potentially unlimited series of dimensional interpretations.


So we move here from the extremely restricted default position of Conventional Mathematics i.e. as 1-dimensional in absolute terms, to an unlimited number of partial relative interpretations, where each number represents a unique dynamic configuration

Therefore from this relative perspective if the interpretation can change as between differing numbers (representing dimensions) then the objective reality then likewise necessarily changes with respect to all these numbers. So from this enhanced dynamic perspective the dimensional notion of number represents perpetual evolution with respect to its very nature.

Now once again, due to the restricted quantitative bias of Conventional Mathematics, this dynamic notion of number evolution is entirely edited out of the picture.

So to give a simple example, when one raises a number 2 to a non-unitary power (i.e. dimension) such as 2, the result is given in a merely reduced quantitative fashion (i.e. as 1-dimensional)!

Thus 2² = 4 (i.e. 4¹).

Now one can easily appreciate, that when seen in geometrical terms, that 2² represents square rather than linear units. However this qualitative change in the nature of units involved is simply ignored in conventional mathematical terms (with a merely reduced quantitative interpretation remaining).


In fact this reduced view is graphically illustrated in the following quote from Alain Connes (from Karl Sabbagh's "Dr. Riemann's Zeros" P. 205).


“It really is a fantastic step to understand that the square of a number - which is just a geometrical square - and the cube, which is just a geometrical cube - can be added together, even though you would say, "But one has dimension the length squared and the other the length cubed" and you would never add things which have different dimensions. So algebra is an amazing achievement, and once you have formulated things in algebraic terms then they take on a life of their own.”


This brings me directly to consideration of the second fundamental set of polarities that govern all mathematical experience, i.e. whole (collective) and part (individual) which in a very direct way determine this key relationship as between quantitative and qualitative.

So, one recognises a number, as for example "2", both individual and collective aspects are necessarily involved (which are quantitative and qualitative with respect to each other)

Thus in external terms, the individual number object "2" that has an actual existence, has no meaning in the absence of the collective number notion of "2" (that potentially applies to all specific instances of "2").

Put another way the recognition of "2", in any specific case, requires the corresponding notion of "twoness" (that collectively apples to all such possible cases).

Then from the corresponding internal perspective, the individual number perception of "2" - again with an actual existence - has no meaning in the absence of the corresponding concept of "2" (i.e. twoness) with a general potential applicability to all possible cases of "2".


So when the individual recognition of "2" is quantitative (in actual terms), the corresponding collective recognition of "2" (or twoness) is - relatively - of a qualitative potential nature.

However, as always in the dynamics of experience, reference frames can switch, with the individual recognition qualitative and the collective recognition now of a quantitative nature. In effect this qualitative recognition corresponds with the ordinal notion of "2" (as 2nd).

Likewise from this perspective, the collective recognition of "2" (as twoness) is now - relatively quantitative (applying to all actual instances of "2").

Therefore in the dynamics of experience, one keeps switching as between both the quantitative and qualitative notions of "2" in individual terms and equally the quantitative and qualitative notions of "2" (as twoness) with respect to both cardinal and ordinal usage. And this happens both externally with respect to objective recognition and internally (with respect to perception and corresponding concept).

And a similar dynamic interaction is involved with respect to the recognition of any specific number.

We then move on to consideration of the general recognition of number.

Once again this will combine both internal (mental) and external (objective) aspects.

And again the general recognition of an individual number integer in a cardinal quantitative manner has no strict meaning in the absence of the collective qualitative notion of number (as "numberness") that potentially applies in all specific cases.

And then when the reference frames switch we attain the individual recognition of that number in a corresponding ordinal manner (which is qualitative in manner), Then - in relative terms - the collective notion of number attains a quantitative interpretation (as applying to all actual numbers).


Now with respect to conventional mathematical interpretation, all these mutually interacting dynamics are short-circuited in a grossly reduced fashion.

Thus as we have already seen,the external/internal interaction is disregarded with numbers viewed absolutely in objective terms (with a corresponding absolute mental interpretation).

Likewise the individual number "2" is interpreted strictly with respect to its quantitative nature, while the general notion of "2" (insofar as it is recognised) is treated merely with respect to actual occurrences (that are likewise interpreted in a merely quantitative manner).

Then it is somewhat similar with respect to the general recognition of a number with both individual and collective aspects treated in a merely reduced quantitative manner.

However if we are to properly understand the key role that the zeta zeros (both Zeta 1 and Zeta 2) play with respect to the number system, we have to inherently appreciate it in a dynamic interactive manner (where both quantitative and qualitative aspects are equally recognised).

Do Numbers Evolve? (1)

On first impression, this might seem to most people as a somewhat ridiculous question.

Indeed the conventional view - which gives great comfort to so many practitioners in our fast changing world - is that number represents the only thing we can rely on to remain absolutely the same. So from this perspective, the prime numbers for example were the same yesterday, today and will forever remain so here and indeed anywhere else in the Universe (where intelligent beings exist to discover them)!

However on closer examination, strictly this conventional view can be convincingly shown to represent but an illusion (which admittedly however in a reduced quantitative sense has proved of enormous benefit).

In physical terms to accurately classify any object we must be able to identify it with a universal class to which it belongs.
To give  a trivial example, to speak unambiguously with respect to a fruit such as a strawberry, we need to be able to define accurately a universal class to which all strawberries belong.

However we will eventually discover that at the margins difficult problems of identification will exist with a certain degree of arbitrariness as to whether a particular example correctly falls into the relevant class. So the boundaries of our definition are necessarily vague and approximate.

Now we might initially think that this problem does not exist in the mathematical world of "abstract" objects that thereby free us from such physical restraints.

However, paradoxically on closer reflection a much greater degree of mystery attaches to the universal class constituting "number" than any physical class (such as strawberries).

So to unambiguously recognise a particular number we should be able to define the universal class of "number". However this is a far more difficult task that one might imagine.

So for example the development of Mathematics has seen a steady increase in the somewhat exotic objects that are universally recognised as numbers.

Initially, number was identified solely with the natural (counting) numbers which are solely positive.. Then gradually, after much resistance their negative counterparts also cam to be included.
A further advance then led to the inclusion of rational fractions (such as 1/2) in the number system. Then the Pythagoreans in investigating the square root of 2 were, to their horror, to discover a new type of irrational number. Such irrational numbers have now been further refined to include both algebraic (such as √2) and transcendental numbers (such as π).

Another major development with respect to the solutions of polynomial equations led to the recognition of imaginary numbers (based on i as the square root of  – 1).

And in more recent times, further developments led to the inclusion of transfinite numbers and a whole new strange class of numbers (based on the primes) referred to as p-adic numbers.


So over the millennia we have seen a remarkable evolution in the objects that are now recognised as legitimately belonging to the number system. Thus it seems to me reasonable to assume that further extension is likely to take place in the future with as yet unknown number objects becoming included.

Thus there is clearly very fuzzy boundaries existing as to what might be considered as number. To put it more bluntly we are unable to properly define what is number and yet attempt to claim an absolute unambiguous identity for every specific number object encountered.

Now though there is no proper (epistemological) justification for such certainty.
So what really characterises - as I hope to presently  demonstrate at length - the apparent absolute nature of mathematical objects such as numbers, is a largely unquestioned mass consensus, which at bottom is geared to the preservation of a considerable illusion regarding their true nature and indeed the true nature of Mathematics generally.

Let me illustrate this now with respect to one of the simplest, most important and best known numbers i.e. "2".

Now again according to the general mathematical consensus, 2 has an absolute rigid identity that can be successfully abstracted from our changing everyday physical world.

This view, expressed in its extreme fashion for example by G. H. Hardy, looks on numbers as eternally existing in some kind of mathematical Heaven (ungoverned by the laws of space and time).

However on closer reflection this view can be shown to be quite untenable.

The starting point here for more authentic understanding is the recognition that Mathematics is intimately bound up with experience. So therefore we start by examining how the recognition of number experientially unfolds.

Now all experience - including of course mathematical - is governed by twin sets of fundamental polarities that dynamically interact.

The first of these relates to external (objective) and internal (mental subjective) polarities.

Therefore the experience of the number "2" entails both an external pole (i.e. as object) and a corresponding internal pole (as mental perception).

So the experience of the number "2" entails a dynamic interaction of both object and perception (which cannot be meaningfully abstracted in absolute manner from each other).

Put another way, strictly speaking a mathematical object such as "2" has no meaning independent of the corresponding mental perception of "2" with both poles in tandem properly constituting an interactive dialogue of number meaning.

In other words all number understanding has a merely relative validity.

So Conventional mathematics is in fact directly based on a reduced interpretation of such experience. Here the two poles are viewed with respect to their absolute separation (though implicitly in experiential terms this is not possible). Thus the number object (in this case"2") is misleadingly given an abstract absolute objective identity.
Interpretation is then misleadingly viewed as simply mirroring in mental terms (again in an absolute manner) this absolute identity.

In this way in conventional mathematical terms, interaction as between opposite poles (external and internal) is thereby completely edited out of the picture (in explicit terms).

This then is misleadingly associated with the considerable illusion that numbers thereby enjoy an absolute rigid identity (unrelated to time).

However because such reductionism is so entrenched in our mathematical thought processes (conditioned now though several millennia) it is extremely difficult to get mathematicians to address this issue.

Even on the rare occasions when I have seen mathematicians seriously question the  basis of such procedures (in philosophical reflection on their discipline), they still seemed in a sense to operate with split personalities, readily accepting all such reductionism (without question) when operating as mathematicians.
And I accept that there is enormous pressure on professional mathematicians (in maintaining the respect of their peers) to operate precisely in this manner.

This is why I have long considered that paradoxically the blunt message that Mathematics (as presently understood) is not in fact fit for purpose can only be properly preached by someone standing outside the profession altogether (while still remaining deeply interested in Mathematics).

A Simple Example

In a recent blog, I suggested that in principle the Erdős–Kac Theorem should have a complementary application with respect to the distribution of prime numbers where the normal (Gaussian) distribution can be used to explain behaviour. To demonstrate this important aspect, we take repeated samples (of same size) within a relatively restricted region of the number system, where changes in the average gap as between primes is so small as to be discounted.

So starting with the n = 1,000,000,000 I took 100 samples of size t = 1000 up to n = 1,000,100,000.

For convenience in identifying the number of primes in each sample, I took them in strict sequence with the accompanying values listed below. As the primes are themselves distributed in a random fashion this would seem permissible in this instance.
Alternatively - at least in theory - 100 repeated random samples of 1000 (with replacement) could be taken within the same range i.e. 100,000,000 to 100,100,000. However in practice this would be more difficult.

So the first row for example represents the 10 samples in sequence from 100,000,000 to 100,010,000 with the last row, for example representing the final 10 sample values from 100,090,000 to 100,100,000.

54	56	57	55	57	61	57	56	47	51
61	55	57	49	43	54	56	43	58	54
51	56	51	49	43	54	56	43	58	54
60	56	55	57	54	52	59	56	51	56
49	43	59	64	55	63	62	53	49	51
63	54	40	54	51	56	52	54	42	57
53	73	55	50	54	53	61	49	52	56
54	59	44	57	50	56	56	53	52	54
57	56	52	54	63	43	54	52	51	48
49	55	57	52	54	52	56	48	56	66

Now in general terms, the mean number of primes in each sample can be approximated as t/log n 

= 1000/18.42 = 54.29 (approx)

This equates well with actual value averaged over the 100 sample results above = 54.11.

Much more problematic however is the provision of a general formula to approximate the standard deviation (for all values of n).

Though I experienced doubts on several occasions with respect to my initial "hunch", repeated empirical testing seems to suggest it as perhaps the simplest and best estimate,

i.e. √{t/(2log n)}

This would give the  standard deviation as 5.21 and compares well enough with the estimated standard deviation (based on the 100 sample values) = 5.46. This does not of course constitute a proof, and indeed a much greater degree of sampling would be required to truly establish it as the most likely estimate.

However in principle by now using the normal distribution, we could estimate the probabilities associated for example of sample values lying within any prescribed distance from the mean (on both sides).

For example we would expect for the above a little in excess of 2/3 of sample values to lie within 1 standard deviation of the mean value.

This would suggest therefore the probability that 2/3 of sample values (for frequency of primes occurring) would lie in the range of 49 - 59 (approx).



Addendum (5/3/2016).  Having returned to this issue in recent days, I feel I can bring more clarity to a situation that I did not really feel had been properly dealt with, first time around.

Though I was hoping that the standard deviation would correspond to √{t/(log n)}, I was led - largely through the empirical evidence of a small sample - to adjust it somewhat to fit the data.

However on reflection this was not warranted. Even just a few stray "outliers" with respect to this data would have a key large influence on the standard deviation. Therefore it was unrealistic to expect that the empirical example would fit in with theoretical explanations.

The  Erdős–Kac Theorem states that if ω(n) is the number of (distinct) prime factors of n, the probability distribution of



is the normal distribution.

In like manner I am suggesting that if  ω(n) is the number of primes in a sample of size t. (i.e. where samples are taken in the region of n) the probabilility distribution of

ω(n) - (t/log t)

√{(t/logt)}

is the normal distribution. 

The Erdős–Kac Theorem would suggest that where the average number of (distinct) primes is approximately 100 (with standard deviation 10), then we would expect just a little more than 2/3 of all factors to lie within 1 standard deviation of 100 either side of the mean (i.e. between 90 and 110).

In like manner if the average mean value of the number of primes in each sample is 100 (where samples are taken in the region of n) then we would likewise expect again that in a little more than 2/3 of samples, the number of primes would lie within 1 standard deviation (i.e. 10) of  100 (i.e. between 90 and 110).

Interesting Connections

We have seen that log n approximates the average gap between prime numbers.

However equally - though less well known - it measures the average amount of natural factors contained by a large number. (In this context natural implies any natural number which is a factor of the number in question!)

In fact Dirichlet proved in 1838 the approximate relationship (for such factors) as

log n + 2γ – 1 = log n +.15443..

So clearly when n is very large, log n provides a very good approximation 

It is also interesting to observe that log n equally approximates the sum of the harmonic series to n (which contains the reciprocals of the natural numbers). The sum (to n) is generally given as log n + γ = log n + .57721...

Once again however when n is very large, log n offers a very good approximation.


Now log log n (Hardy-Ramanujan Theorem) offers an approximation of the average number of (distinct) prime factors contained by a number (when n is very large)

It is tempting therefore to presume that, equally as with log n, a complementary type relationship might exist with respect to the average gap as between natural numbers.

After some consideration, I came up with the notion of spacing as between composite natural numbers composed of non-repeating prime factors. So for example 6 (2 * 3) and 10 (2  *5) would represent appropriate examples. However 4 (2 * 2) and 8 (2 * 2 * 2) and  9 (3 * 3) would be excluded. All single primes however could be included.  

I am offering no proof of this and have only had the time to carry out limited empirical testing of the proposal. However it would represent an interesting hypothesis to test.

It is also once more interesting to observe that just as log n approximates the sum (to n) of the reciprocals of the natural numbers, that equally log log n approximates the sum (to n) of the reciprocals of the prime numbers.

Again the more correct approximation is given as log log n + B (where B is Merten's number = .261497...). However for large n the simpler expression i.e. log log n would offer a good approximation!


It is also fascinating to note that log n/log log n measures the ratio of the average amount of natural number to (distinct) prime factors for a large number.

Equally it measures the ratio of the sum of the reciprocals of natural numbers (to n) to the corresponding sum of reciprocals of the prime numbers.

Also it has been proven that if n is primorial - i.e example 2 * 3 * 5 * ...* n, that

log n/log log n approximates the number of factors in n (for very large n).

Once again log log n measures the average number of (distinct) prime factors in n.

It would therefore be fascinating to obtain a corresponding measurement for the total number of prime factors (allowing for repetition).

So for example 2 and 3 are the (distinct) prime factors of 24. However 2, 2, 2 and 3 represents the prime composition of 24 (allowing for repetition).

Now once again - based on a limited amount of empirical testing - I offer the expression (log log n)²

as the simple appropriate expression in this case.

Finally, log n measures the average amount of natural factors.

However for example if we say that 24 contains 1, 2, 3, 4,, 6, 8, 12, and 24 as factors (There is always some arbitrariness as whether to include 1 and the number itself as factors! However for very large n it does not significantly affect approximation results!)

This would therefore represent 8 factors.

However, these factors could be uniquely expressed as 1 * 24, 2 * 12, 3 * 8 and 4 * 6.

So 8 is now reduced to 4.

So if we represent factors in this latter fashion (as unique combinations of 2) the corresponding expression for the average number would be log n/2.


Addendum 21/3/16

I have since discovered - through access to detailed tables listing the prime factors of all natural nos. to 10¹⁵ - that the above contention that the average number of prime factors (including those that repeat) would ~ (log log n)² is incorrect.

It seemed to me to be be intuitively likely (without initial access to the prime factor tables) that the natural numbers with recurring prime factors would steadily increase, as a proportion of all numbers, as n increased.

However this is is not the case with a constant relationship apparently being maintained throughout the number system.

Therefore the average frequency of all the prime factors of a number (i.e. where factors can recur) can be given as k(log log n) where k is a constant.

Now this constant would appear to work out approximately at just a little in excess of 1.2.

I have speculated in recent blog entries that the value the value of k ~ (1 + √2)/2.

More on Erdős–Kac Theorem

One of the valuable features of the Erdős–Kac Theorem is that it enables us to calculate - using the normal distribution - probability estimates for the various numbers of (distinct) primes that might be associated with a given large number.

Now it is amazing how large these numbers quickly get. For example as the Wikipedia reference makes clear, numbers with 10,000 digits would require (on average) just 10 (distinct) primes for their construction.Thus if n is such a number (with 10,000 digits) the mean average i.e. log log n = 10 (approx) and the standard deviation =  √(log log n) which is just slightly in excess of 3.

We can then say that just over 2/3 (i.e 68%) of such numbers would fall within one standard deviation either side of this average which would entail that about 7 - 13 (distinct) primes would be involved in the construction of a 10,000 digit number!

We could equally use the normal distribution to calculate the probability of 1, 2, 3, 4, .....k (distinct) primes comprising a such a number (where k represents the maximum number of distinct primes that could be involved).

It is important to realise however that as we keep ascending the number scale that distinctive normal distributions would be used.

Thus for example the normal distribution that would describe the typical behaviour of a 100,000 digit number for example, would necessarily be of a different shape to that for the corresponding 10,000 digit number!

In general terms as the size of the standard deviation progressively falls in percentage size (relative to the mean), we can therefore expect - ever stronger clustering of values for the number of (distinct) primes around a central average value as the size of n increases..

I suggested in the last entry that a complementary Erdős–Kac (Type 1) Theorem should equally exist.

Once again we can have two complementary opposite notions of the relationship of the primes to the natural numbers.

In Type 1 terms - which represents the standard approach - the primes are viewed in an (individual) manner as having a random existence with respect to the (collective) natural number system.

However in Type 2 terms - which relates directly to the Erdős–Kac Theorem - the primes are now viewed in a (collective) manner as factors with respect to an (individual) natural number!

As Type 1 and Type 2 are directly complementary (mutually implying each other), this suggests that a corresponding Type 1 Erdős–Kac Theorem should apply (complementing the recognised Type 2 version).

Misleadingly I suggested that this alternative distribution might apply to the gaps as between the primes. However following some empirical analysis, it became quickly apparent to me that the measurement of the gaps as between primes (with respect to a typical average gap given as log n) would not correspond to a normal distribution.

Paradoxically, though the individual primes are indeed random with respect to the (collective) natural number system, the actual gaps as between primes are highly ordered. This highlights again the fact that notions of randomness and order can only be properly understood in a dynamic interactive context, where they mutually imply each other.

Thus the very reason why the (individual) primes can be random (in this Type 1 manner) is due to the perfect order with respect to the respective gaps between primes. Without such complementary order with respect to the distance between successive primes, the primes could not preserve their random nature!

So the complementary Erdős–Kac (Type 1) Theorem does not apply directly to the gaps as between primes, but rather the frequency of primes.

As we know this frequency of primes can be simply approximated as n/log n (which steadily improves in relative terms as n increases).

Thus once more when n is very large, we can take successive similar sized random samples of primes (that are large in absolute terms though very small relative to n).

The actual number of primes occurring in each case will tend to cluster around an average value with deviations occurring with respect to the average value.

Now the contention is that these deviations from the average value will be normally distributed!

Thus through knowledge of the standard deviation, the distribution can be standardised so that in principle we can predict the probability percentages for actual number of primes occurring within any distance from of the mean.

The mean average mean value for sample values will be given as t/log n.

The standard deviation seems to be somewhat less than √(t/log n) and approximated roughly by √(t/2 log n).

So to illustrate, in the region of n = 9.000,000, log n = 16!.

Therefore if we were to take repeated samples of say 2,000 in this region of n, we would expect the mean average number of primes to be about 2,000/16 = 125.

Now the actual values of primes occurring will deviate from this average, with the standard deviation roughly √(2000/(2 * 16) = 7.9 (rounded to 8).

Therefore, we could use the normal distribution to estimate that about 2/3 (68%) of the recorded actual number of primes in each sample would lie between 116 and 134.

Needless to say, I am merely suggesting here that a complementary Erdős–Kac (Type 1) Theorem in principle should apply to the primes, that can be precisely represented (in a relative manner) by the normal distribution.

However I have not suggested here the proof nor indeed clarification as to the precise standard deviation.



Addendum (8/10/2016) As mentioned in a later entry I have since accepted that the standard deviation should be √(t//log n).