When something appears to be quite complex, it can be quite exhilarating to find that there is a very simple basis for that complexity. Although this conjecture may never be solved, never-the-less it has secrets that can be uncovered.

The following graph represents the positive integers from 2 to 1,000,000 on the x Axis and the number of pairs of prime numbers on the y Axis that can form each of those integers.

When I first looked at this graph, I didn’t even attempt to think that there could be an understanding possible. Sometimes it is only in hindsight that you see something that evaded you previously.

The patterns of bands suggest a strong behaviour but quite what, has eluded everybody until now. Having discovered and researched the Prime Number Channels [1] which reveals the different patterns of alignments and how they should affect variability of prime numbers pairs for different values of even integers, I realised that there was an excellent chance that this had something to do with the bands.

Being more a scientist than a mathematician, I used the tried and true scientific method of trying to put forward a hypothesis, that numbers which had lower levels of alignments of Prime Number Channels should have fewer prime number pairs also.

My belief was that the bands were in four groups, each group identified by the number of alignments of Prime Number Channels per consecutive block of 30 numbers. The pattern suggested to me that the scatter plot could be broken down into four regions, each region having a dense band followed by a faint band, then a smaller less-dense band followed by a fainter band.

First, I created a database of the first 500,000 prime numbers excluding 3 and 5 which are immaterial for integers larger than 30. Next, I created a database of the first 1,000,000 even integers and then paired all different combinations of prime numbers and recorded where they all fell in order to count the pairs that each number had..

The critical stage was to record the type of even integer as either 3, 4, 6, or 8, being the possible number of Prime Number Channel alignments per block of 30 consecutive numbers. The relevance of these channels had been downplayed by the mathematicians I had shown them to and I had realised that in order to gain recognition you had to go a long way and show a lot of things before anyone would take you seriously. This meant that all the research I did was mostly in isolation as I had long given up hope that most mathematicians would appreciate the importance of simplicity. Although mathematics is in the science faculty of a university, the desire to seek proofs ahead of understanding makes them lesser scientists because a proof doesn’t always provide the deepest understanding of a problem (FLT is the best example of a problem supposedly proven but without any clarity of the mechanisms of the problem). A scientist notices an observation, formulates a hypothesis, designs an experiment to test the hypothesis and then looks at the results to see if anything has been gained or if further hypotheses are called for.

Scientists seek reasons for observations while mathematicians seek proofs.

Now I had the database to start making selections from. First I extracted the type 3 even integers. Plotting them reproduced the bottom quarter. Type 4 even integers gave me the next quarter, Type 6 even integers the third and type 8 even integers, the last quarter. My broad hypothesis was correct!

Although elated, I assumed that it was probably going to be too difficult to work out the lesser bands. Anyway, I decided to proceed and play around with a few selections of the type 3 even integers. I guessed that 7 and possibly 11, and 13 were something to do with it so I extracted numbers either side of multiples of 7, 11 or 13 and plotted them. All the plots appeared to be fainter versions of the original plots and I decided that it was probably going to be so difficult, I would leave to someone else to figure out…

After a restless night I decided I wasn’t going to let someone else discover something simple by standing on my coat tails.

Upon closer inspection of type 3 numbers and the variation of numbers of prime numbers, I did some simple divisions and discovered that the numbers that had the highest number of prime pairs, happened to be divisible by either 7, or a combination of low prime numbers.

My next extraction then was even integers which were divisible by 7 and Voila! I had something that was very, very close to the top half of the bands for type 3 numbers.

When I excluded numbers that were not divisible by 7, I found the bottom band reappearing.

That was when I realised that when a number is divisible by low prime numbers, it means that other prime numbers are included as factors of all the divisions of that even integer. In other words, the opportunities for prime numbers lining up are increased. Conversely powers of 2 are therefore likely to have fewer pairs of primes.

It is exceedingly clear that the understanding Prime Number Channels and patterns of alignment of these channels gives a deeper insight than anything done by way of traditional proof searching. Knowing for example that all even integers can be formed by at most 6 prime numbers summed together doesn’t provide any deep meaning although it might provide intellectual rewards.

There is a prevalent belief in mathematics that no complex problem can be solved with a simple solution as someone else would have found them. Chaos Theory demonstrates in many examples that complex things are caused by the interaction of simple things but finding those simple things is actually quite complex. Simple solutions does not equate to simply found. It’s simple to find complex things as complex things avoid deep understanding.

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*References*

[1] Paper on Prime Number Channels

[2] Paper on Goldbach’s Conjecture Density Bands

*Phil Jackson runs his own small software company, Concept Patterns Limited. He can also be found on the Simplicity Instinct website.*

Mathematicians had all but given up hope of ever understanding prime numbers. The last time there was a brief flurry of excitement was when it was discovered that the periodicity of zeros from the Riemann-Zeta function closely matched the spacing of energy levels in high-Z nuclei [1]. Prior to that, Gauss and others [2] had showed a relationship between Pi, Logs and prime number density. You then have to go back to Euclid some 2,300 years ago for his proof of an infinity of prime numbers [3].

Number theory sits at the centre of the mathematics world and despite a massive effort to understand prime numbers, little is known about them. A number of important conjectures (Riemann’s Hypothesis [4], Goldbach’s Strong Conjecture [5] and others) lie unsolved and perhaps unsolvable.

The mathematicians working on prime numbers are very smart people who are able to manipulate a group of abstract concepts in their heads which leaves myself and others wondering which world they come from. But having adopted a unique problem-solving method in my own software company that proved highly successful, I decided to dabble in mathematics as an exercise to see if this approach to solving complex problems could find answers when other approaches were clearly not delivering. What ensued was an adventure of discovery and obstacles that lasted over six years.

The first project I started on was to look at the Four-Colour Map Theorem and discovered a way of simplifying it but failed to convince a local mathematician and a professor from my old university of Otago of its relevance. This was my first experience of mathematicians discounting simplicity and this was repeated many times. Looking back at this problem after six years made me realise that there is an even better was of simplifying that should lead to a simple proof but I sense that it will be ignored.

Undeterred, I started working on Fermat’s Last Theorem and Goldbach’s Conjecture and that is when I observed a pattern in the pairs of primes that combine to form all even integers according to this latter conjecture. Driven by a need to understand this observation, I spent almost two years getting to grips with this before I demonstrated this to a local professor and a senior colleague. They were intrigued but again dismissed it as not relevant, believing that the French mathematician Dirichlet already covered this.

At this stage most people would have accepted the advice of those considered to be much more knowledgeable but my trusted intuition said they were wrong. History will show that I was right to persevere with two papers on prime numbers now published on the internet [6,7] that reveal a number of facts about numbers in general and prime numbers and all are based on simple mechanisms requiring nothing more than high school mathematics to understand.

In my first paper, I discovered that multiples of 3 and 5 form a palindromic pattern, leaving all higher prime numbers and multiples of those prime numbers, evenly distributed between eight channels I called the Prime Number Channels. This provides a basis for understanding Goldbach’s Conjecture in terms of why some even integers are more likely to have more pairs of primes than others, something that over 200 years of research failed to produce.

In my second paper, I describe simple attributes of prime numbers and show a remarkable graph that presents a powerful and convincing argument that groups of prime numbers occur between boundaries.

My scope of research also included looking at Riemann’s Hypothesis and the nature of the multiple of 3 and 5 strongly suggested that this problem was probably not worth pursuing as it didn’t take into account that all other prime numbers need to be treated as a separate group. It also made me wonder why researchers were using a manipulation of logic to prove this hypothesis when in reality they were trying to prove that the behaviour of prime numbers, which is unknown, matches this model.

In my papers I make many suggestions for further research and want to start a new period of interest in prime numbers. As to my secret method of problem solving and the reasons why I believe that mathematicians have failed where I have succeeded, you’ll have to wait until my book ’Simplicity for a Competitive Advantage’ is published. The draft is ready and and I am looking for a publisher to pick it up.

My papers are not written in an orthodox terminology, a result of my limited knowledge of mathematics convention, but in the end I wanted to write them in a way that opened my ideas to an audience that is wider than mathematicians. The mechanisms of number interaction were not designed by mathematicians and belong to everybody. Finally, I have held back two papers which I will release in the future and these will be sure to create some surprises.

*References*

1. The stability of electron orbital shells based on a model of the Riemann-zeta function.

2. Prime Number Theorem

3. Euclid’s Proof of an Infinite Number of Primes

4. Riemann’s Hypothesis

5. Goldbach Conjecture

6. Paper 1

7. Paper 2