The Prime Numbers Sequence: a 2000+ year old enigma resolved

Primes_books

Here’s an excerpt from my online ‘paper’, The Primal Code, that’s been around since 2006.

It has been read by various professors and teachers of mathematics, scientists and others, and has not been refuted, despite open invitations to do just that. The problem for these learned people seems to be that The Primal Code starts from a broader perspective than that of current conventional mathematical thinking, and only later homes in on the numbers themselves to resolve the mystery.

With most mathematicians I’ve encountered, this has resulted in a judgement that it’s ‘not proper mathematics’, and therefore not worthy of serious consideration, even though it delivers a sound, consistent and clear resolution to a previously unresolved enigma.

It works like this. Three simple axiomatic principles are posited which govern how the cosmos originated and continues to function. As such, these are fundamental to all thinking, including mathematical thinking, in the broad sense of its Greek root word ‘manthanein’ which means ‘to learn’. The three principles are outlined in Notes from the Threshold, available online as a free download from www.pathwayinitiatives.co.uk.

The third of these principles is resonance which connects Pythagoras’ Harmonic Series with Euler’s discovery of (a) the basic mathematical zeta function and (b) how the prime numbers connect the functions of multiplication and addition. So the work of Pythagoras and Euler together link resonance with the primes.

The Primal Code is not concerned with calculating probabilities or mathematical proofs, since these necessarily involve various questionable assumptions integral to the discipline as it currently stands. Instead, it offers a concise and logically consistent line of thinking which delivers the required numbers in a coherent, clear and systematic way. And just for clarification, this work is not based on any religious or mystical beliefs. All it requires is an open minded, non-dogmatic attitude and a little perseverance with what at first may seem an unfamiliar way of thinking.

                                                                              

The Primal Code (V9.06)

– reveals a simple number sequence

which has mystified mathematicians for centuries,

and which records how, out of chaos,

the cosmos began and continues to be recreated

– resolves the three oldest and greatest enigmas

in mathematics:

The Prime Numbers Sequence (2000+ years)

The Riemann Hypothesis (1859)

The Goldbach Conjecture (1742)

– offers an elegantly simple resolution to

Fermat’s Last Conjecture (1637)

(misleadingly called a ‘theorem’ because it lacks any backup workings)

© Pathway (Initiatives) Ltd 2006-2014

http://www.pathwayinitiatives.co.uk              info@pathwayinitiatives.co.uk

 

What mathematicians & scientists have said

…The primes represent the most tantalising enigma in the pursuit of human knowledge…”

Cover blurb of The Music of the Primes by UK professor of mathematics, Marcus du Sautoy (2004).

“Upon looking at these numbers, one has the feeling of being in the presence of one of the inexplicable secrets of creation.”

Mathematician Prof. Don Zagier speaking of the primes, quoted in The Mathematical Experience by Philip J Davis and Reuben Hersh (1981).

Prime Numbers – The Most Mysterious Figures in Math.

Title of a 2005 book by mathematician David Wells.

God’s Secret Formuladeciphering the riddle of the universe and the prime number code.

Title of a1997 non-fiction book by scientist Peter Plichta.

Prime Obsession …the Greatest Unsolved Problem in Mathematics”

Title of a 2003 book about the prime numbers by mathematician John Derbyshire.

These few quotes are included simply to illustrate the continuing fascination after many centuries, particularly of mathematicians and scientists, with the prime numbers. The books reveal how each author seemed intuitively to sense something very special which inspired his book.

Yet none was able to grasp what actually makes the prime numbers so significant.

The primes are also central to the Riemann Hypothesis and the Goldbach Conjecture, two other longstanding and unresolved conundrums of mathematics.

Albert Einstein made the wise observation that “No problem can be solved from the same level of consciousness that created it”. This one can only be resolved by starting out from universal cosmic principles beyond mathematics, and then homing in on the world of numbers, or working from those numbers to the principles.

______________________________

Prologue

Central to this work is the idea that the very sequence of the prime numbers constitutes a Primal Code of values which is fundamental to understanding how the cosmos was and continues to be formed. In this bigger context, a universal coherence can be appreciated, certain mathematical problems, previously considered unresolvable, can be resolved and various special numbers, called ‘irrational’ by mathematicians, can be seen as rational.

Since the approach taken here is by no means an orthodox mathematical one, it does involve questioning some perhaps previously unquestioned assumptions. Embedded in normal mathematical practice for many years, these would seem to have been contributing to the problems all along.

In the 1930s, mathematician, Kurt Godel, established that there are statements which cannot be proved true or untrue within the axioms of a mathematical system. For a mathematical ‘proof’ only has meaning within the limited definitions, rules and conventions of the language of mathematics. So meaning cannot be found in numbers themselves, although patterns of order amongst them obviously can and may imply meanings.

It’s also assumed in this work that there is a coherent order behind what’s normally called randomness, a concept that covers up the human inability to handle certain degrees of complexity arising out of the basically simple cosmic principles underlying that order.

An understanding of the pattern and meaning of the prime numbers sequence has eluded mathematicians for over 2000 years, and since 1859 no one has been able to prove or disprove the Riemann Hypothesis. The Goldbach Conjecture, about a self-evident fact regarding even numbers and the primes, has likewise remained unproven since 1742.

So questions inevitably arise concerning the assumptions behind the thinking of all those striving to resolve these three related enigmas. That is to say, these problems have to be viewed in context from outside the mathematician’s mindset, from outside the box, with questions more of a psychological than a strictly mathematical nature.

Assumptions of Threshold Mathematics

1. Universal Consciousness (UC) is the original animating principle of the void/vacuum of no-thing-ness, an idea beyond the limited capacity of individual, incarnate human consciousness to comprehend fully. UC is axiomatic to this work since it’s what accounts for humans, individually and collectively, being able to think about such abstract subjects as mathematics.

2. The second and third universal principles, polarity and resonance, account for the coherent order that prevails throughout the cosmos, behind all superficial disorder.

3. Zero has two complementary, polar opposite meanings, both of which carry the sense of a threshold which itself has no substance or numerical value. Big zero is the periphery where finite ‘somethingness’ and the infinite ‘nothingness’ interface. Little zero, as a number symbol, represents those points, gaps or spaces which serve as portals into the same infinite nothingness. That theme will be developed later in this work. Beyond zero is the so called ‘negative’ realm of potential/virtual values, the great void of no-thing-ness, aka the fullness of pure energy.

4. The number One serves as both the biggest and smallest number: the all-inclusive, total unity and its polar opposite, the exclusive, single unit.

1. The Threshold Perspective

Two Greek words: khaos (chaos) implies a potential, pre-physical order in the void, womb, matrix, cauldron (the negative), while kosmos (cosmos) means a worldly order (the positive).

The cosmos/chaos cycle

This refers to the continuous cyclical process of manifestation from the pre-physical state of chaos into the physical-material state of cosmos, followed by redemption and so on.

1. It all starts with UC, the void, full of potential, within which is created a focal point, a singularity. That point becomes the centre around which an expanded sphere instantly forms, ie a pre-physical sphere of potential manifestation, a oneness, an order, a cosmos.

2. The symmetry of this sphere is distorted like a bubble by uneven ‘external’ and ‘internal’ forces. The resulting asymmetry accounts for the various natural, archetypal forms and shapes. Consideration of the primal forces involved is beyond the scope of this work.

3. As universal consciousness tends towards re-establishing its original unity state, a rhythmical, pulsating dynamic of two polar opposite forces arises between periphery and centre, inwards/outwards, an order of two-ness within the one cosmos. This dynamic 2dimensionality, D2D, is a key concept regarding the currently evolving mathematics of the Threshold.

4. The interaction between the polar opposite forces of (a) inward contraction (gravity creating density) and (b) outward expansion (levity creating rarity) generates a primal, pre-physical potential, a dynamic, ie basic, raw ENERGY.

5. The interface, the threshold where these two forces meet and balance out, establishes three-ness from the two forces in the one cosmos. This threshold can be imagined as a spherical standing wave, transient but enduring so long as the two polar opposite forces maintain their dynamic, rhythmic state of balance.

6. The threshold standing wave, a concentric sub-sphere within the original sphere of potential manifestation (the cosmos), also constitutes a resonant cavity.

7. Within it, repetitively, further sub-spheres are formed in accordance with the principle of harmonic resonance, which the acoustical Harmonic Series of Pythagoras demonstrates.

8. Towards the centre point, the increasing density of concentric sub-spheres impedes the rhythmically pulsating energy until eventually some becomes trapped, frozen or ‘woven’ between the densely packed sub-spheres closest to the centre.

9. Here matter is created, ‘woven’ through the interplay of the inward/outward movement (weft) and the standing wave inertia (warp). The process passes through the four states of matter – from the rarest, transformative, fiery plasma, through the gaseous and liquid, to the densest inert solid. These four states correspond to the four traditional archetypal elements of fire, air, water and earth.

10. When the original universal consciousness, as primal energy, cools, contracts and sinks (‘descends’) into dense solid matter, disintegration and fragmentation occur. Discrete bits of material substance form within the previously continuous fluidity, like lava.

11. Meanwhile, individualised, earthly human beings have been evolving through a pre-physical, aethereal state into physical-material embodiment, and begin to use, compare, standardise and measure material objects in an embryonic mathematical way.

12. The now individual, point-centred consciousness of incarnate humanity, deeply embedded in matter, gradually evolves the capacity for detached, abstract thinking, ie able to disconnect from its immediate physical circumstances. That enables a way of quantifying to develop, applicable to all aspects of the physical-material world: an equal measuring unit (EMU) system of standardised quantities, and an EMU number system.

This very useful but artificial device does not truly reflect the dynamic continuity of Nature and the cosmos. It requires a kind of suspension of disbelief, like the cinematic experience of ‘seeing’ a series of still pictures as if moving and alive. It has, however, now long dominated and distorted humanity’s conception of what might be ‘real’, true or factual.

One pathological consequence of this is the current epidemic of sterile, ‘objective’, dead (static) statistical thinking. This excludes the living dynamics of will and feeling, ie pro-active consciousness, from all calculating, analysis and projecting into the future.

13. In the 20th century, the reversal from the descent into matter and materialism towards the ascent out of matter began to be reflected in physical science – especially in quantum physics. The shift of emphasis, in this bi-polar cosmos, has significantly affected human consciousness as it continues to progress and evolve in both directions, as follows:

(a) Further inwards leads ‘through’ the centre, the inner infinitude or vanishing point of density and smallness. That means penetrating beyond the atom (the Greek word ‘atom’ means an indivisible entity), and beyond the so called quantum level (‘quantum’ means the smallest discrete quantity of physical energy). This ‘passing through’ the cosmic centre point results in a ‘re-emergence’ into the non-quantifiable, continuous pre-physical realm.

Physical science, having not yet realised this, still uses the materialistic terminology of fragmented, separate particles etc in its endless categorising and quantifying.

(b) Further ‘outwards’ leads directly to the peripheral infinitude of rarity and vastness, ie into the pre-physical realm and the unlimited void of UC. Astrophysicists and cosmologists search in this direction, but also still seem locked into the terminology of the physical-material world – while allowing themselves a little imaginative poetic license here and there, such as in curved space, event horizons, black holes, dark energy and so on.

A new mathematical language has long been required to deal with the pre-physical realm, given it’s qualitative difference from the physical-material realm from which the EMU system emerged. And, as will become clear later in this work, certain improvements to the currently conventional number system are required.

To recapitulate, the continuity of the cosmic formative process can be summarised as:

-1: From the unlimited, negative nothingness/void/fullness of Universal Consciousness beyond…

0: the zero (non-dimensional) Threshold, emerges as…

1: the oneness of a pre-physical sphere of potential cosmic manifestation, forms

around a focal point (one-dimensional), the periphery and centre together creating…

2: the dynamic two-dimensional two-ness of the primal polar forces, gravity and levity,

out of which arises a pulsating matrix/medium of primal, potential energy, in which…

3: three-ness is present at the Threshold between the two forces in the one manifested cosmos.

2. The Prime Numbers Enigma (2000+ yrs)

In the pre-physical realm there are no discrete, separated parts – only an ever changing continuity of different qualities, forces, tendencies etc. This would seem to be what has confounded many who have delved into the so called ‘quantum’ world. So, mathematically, you can start from one as a whole inclusive unity or from one as a partial exclusive unit, all within the greater nothingness.

To divide up the wholeness, the unity, into a number of equal portions is to multiply the quantity of units by that same number. So the universal principle of polarity finds mathematical expression in, for example, the reciprocity of division and multiplication.

Six-ness

The process of manifestation into many-ness is one half of a two part cyclical process, the other half being the polar opposite complementary process of redemption back into the original state of oneness/nothingness – ie the general idea of entropy. So, one whole cycle consists of two phases (inward and outward), each of which consists of three stages. That makes six stages per cycle.

This can be considered multiplication: oneness producing two-ness producing three-ness, or addition: a state of simultaneous oneness plus two-ness plus three-ness.

Whether 1, 2 and 3 are multiplied or added together, the result is 6, product or sum.

This special property is unique to the number 6 which is thus known as a perfect number.

Pythagoras’ Harmonic Series connects resonance with the mathematics of ratios, while Euler’s discovery of the basic mathematical zeta function coincides with this.

Euler’s Product also demonstrates how the prime numbers connect multiplication and addition. So the work of Pythagoras and Euler together link resonance with the primes.

The Primal Code begins to emerge in two arithmetic progressions, each with the common difference of 6, which express mathematically the inward and outward rhythmical and resonating pulsation of the cosmos. One starts from -1 (representing the pre-physical potential, ie chaos) and the other starts from +1 (representing the manifest physical cosmos). Thus the formula:

Tn = 6n ± 1

As these series progress, each new number/resonant cavity that is a multiple of (ie resonates with) an already established prime is automatically subsumed by it, has no independent resonant identity of its own and so is cancelled out, eliminated.

That leaves only those RCs that resonate solely with the original one cosmos and those numbers which are only divisible by one, ie the prime numbers.

Artificially merging the two arithmetic progressions gives us the familiar sequence of prime numbers, which previously appeared to be ‘irrational’. These are, in essence, a series of ordinals to which the cardinal prime numbers correspond in the Primal Code.

A double helix explains the ‘twins’

                                                            Primes seq:Res symb

When the two progressions, from -1 and +1, are displayed as a double helix, the two interweaving spirals, pass closely at every count of 6. This graphically de-mystifies the previously enigmatic phenomenon of the twin primes, such as 11 and 13 or 29 and 31. (See page 8)

To summarise, briefly, the formative process:

Pulsation between periphery and centre of the original cosmic sphere/resonant cavity creates standing waves which form, within that original one sphere, a series of concentric

sub-spheres/resonant cavities in resonance solely with the original.

That series – in order of decreasing magnitude and faster rhythmic resonance –

corresponds to the acoustical/musical Harmonic Series of Pythagoras.

The Harmonic Series coincides with Euler’s Zeta Function (when the exponent x = 1).

The Zeta Function (when x=1) is an infinite series.

The Harmonic Series is, therefore, an infinite sequence of resonances, its fundamental

(lowest) tone being that of the original one resonant cavity.

The Primes, as cardinal numbers, are mathematically ‘resonant’ only with number one,

while all other EMU numbers can be reduced down to combinations of primes.

Thus the Primal Code has a dual function as:

(a) a series of ordinals (1st, 2nd, 3rd…) corresponding to the infinite sequence of cavities which are resonant only with the original one, and which can be said to represent primal values, and

(b) those EMU cardinal prime numbers only divisible by (ie resonant with) the number one.

‘Generating’ primes and ‘predicting’ the next one

The expression ‘generating primes’ can now be understood as having two meanings:

1. The forming of the cosmos in stages, expressed as a series of ordinal values, and

2. Mathematically calculating a specific series of EMU cardinal numbers.

The abstract mathematical exercise of ‘predicting the next prime’ is thus a question of:

(a) the limits of current computational technology, and

(b) how much time and resources anyone is prepared to devote to this infinite quest.

A practical working definition of the primes

The primes, expressed as EMU cardinal numbers, represent an ordinal scale of markers of the cosmic formative process – from the potential pre-physical realm to the realm of actual physical forms – expressing the principles of polarity and harmonic resonance.

So the primes can be understood as a primal cosmic number system, a Primal Code. However, when seen only from the physical-material perspective of the EMU number system, no logical order is apparent. Hence their 2000+ year ‘enigma’ status.

The fact that this account is not wholly reducible down to a concise formula, algorithm or a formal ‘proof’ affirms Godel’s Incompleteness principle and emphasises that the significance of the primes extends beyond the limits of current mathematics. One idea for extending mathematics beyond this present limitation would be to introduce a new mathematical resonance symbol (ie stating ‘is resonant with’), as proposed in Threshold Mathematics.

Also, to ignore the subtle, underlying pattern and the greater cosmic context, in favour of a narrow, arithmetic definition of what constitutes a prime number, would be to repeat the failure over the last 2000+ years to appreciate the significance of the natural order of the primes.

Thus the enigma of the distribution of the prime numbers is resolved by the Primal Code.

____________________________________________

3. The Threshold Rationale of ‘Irrational’ Numbers

From the Threshold perspective, certain so called irrational numbers can be understood as rational Threshold numbers, since they regulate or scale the transformations between the pre-physical and the physical-material realms – serving as a kind of constant exchange rate.

They represent constant scaling values of proportion or ratio between the two realms which, being qualitatively different, function according to different ratio-nales.

The fluid, pre-physical realm is not solid and fragmented into separate, discrete, quantifiable EMU bits. Which is why, from the physical perspective, any crossover to or from the pre-physical realm seems irrational, ie has no precisely calculable ratio... (continued)

The whole document, The Primal Code, is available free as a download from http://www.pathwayinitiatives.co.uk

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