Unexpected Prime Number Breakthrough

By Guest Author 19/07/2010 9


By Phil Jackson

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

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


9 Responses to “Unexpected Prime Number Breakthrough”

  • I read your first paper and I didn’t find it that compelling (I also think your statement that they are ‘published’ is a little misleading). Your palindromic pattern simply occurs because 3 and 5 divide 30/2. In fact for any integer n if you pick sets of integers that divide n/2 you will also find such palindromic patterns. Your table of channel multiplication is simply the Cayley table of the abelian group of the set Z*_30 under multiplication. You got this because you removed factors of 2, 3 and 5 and thus are left with all and only the numbers that are coprime with Z*_30. After that as far as I can tell you simply elucidate a number of properties common to such groups. Incidentally a number of times you seem to use a table as ‘proof’ of your idea. This is not a valid argument in maths (unless you actually can enumerate all possibilities. For a historical example as to why look up Fermat primes). You may want to consider doing the proof by induction.

  • The papers are well written. Yes, I give you your propers, for making effort, where many a Grad student claims they are beyond us.

    However, in math, so many things, are just beyond most people. It is hard to believe, any simple pattern exists to primes. They are just that way. (from http://simplicityinstinct.weebly.com/1/post/2010/07/introduction.html )

    Sure, the second gentleman has no doubt skunked us both. Although, I would also suggest looking at Twin Primes, too.

    I would simply want to say, don’t stop trying. Do what is fun. Be realistic, keep your best insights a bit longer, look for more contradictions, the best book for proofs – Mr. Polya’s classic for starting, beyond this – find a math teacher who is NOT; terse, obtuse, pedantic, arrogant, or willing to show the basis / fundamentals. (That will be pretty hard. Math proofs are written without clarity, if you ask me. ie, That is the terse word.)

    So, keep up the good work. Check out pseudo-primes, Fermat’s Theorem is done. Although, the solver “refuses to accept” his million dollar award, too. Strange hunnn? Make up your own form of primes?

    Rock On – Rob

  • AS for you happy evil slosh,

    I would like to see you, produce a proof, there is no solution for a formula to produce all primes. (Excluding Euclid’s Algorithm). Then, write it, as nicely as Mr. Jackson has here.

    Should be simple, don’t you think? (Yes, I’m being sarcastic.)

    Rob

  • You misunderstand me robparn.

    When I used ‘simply’ (although after posting I realised I used it more than I found aesthetically pleasing, unfortunately there is no edit/delete on comments that I can see) I meant as in ‘your idea is almost fully encapsulated under this subject heading’, not that it was a simple idea. When entering a field I’ve found one of the major battles is determining what certain ideas are called, unless you’re up with the literature it isn’t always obvious.

    As for your comment on Fermat’s theorem I’m going to just make sure you realise Fermat’s theorem isn’t related to Fermat primes (other than the inclusion of the mathematician, Fermat). Further I don’t think you’re right that Wiles turned down money for his proof of Fermat’s theorem, in particular I don’t recall a reward for it’s proof ever being offered. Maybe you are thinking of Perelman who proved the Poincare conjecture?

  • When I first showed the prime number channels minus the prime number distribution and equations that generate non-prime numbers, a senior researcher thought that Dirichlet’s work should cover all individual “channels” and therefore finding a set of “channels” shouldn’t be considered as anything unusual.

    Cayley tables are similar to Dirichlet’s “channels” in the sense that you can take a set of numbers and produce a table with standard characteristics. By extrapolation the assumption is made that all tables are similar as was made with Dirichlet’s work for all channels.

    My argument is that in blocks of 30 this table is of significance because it is the only table that excludes multiples of 3 and 5 (other than larger blocks of 30 that is). None of the other tables provide the same insight into Goldbachs Conjecture because they don’t reveal the probable variability of prime-pair matches at different positions.

    The equations for each channel provides a deeper level of understanding of the mechanisms involved and it may be that someone may look at these and see something that I have missed.

    Finally in respect to Riemann’s Hypothesis would someone like to demonstrate that the problem cannot be distilled down to being described as proving that the model landscape (as it is often described as) has to be shown as matching the behaviour of prime numbers (which is unknown). If this cannot be done then that makes it a problem that can only be solved by understanding prime numbers and that has not been the avenue taken to date as far as I am aware.

  • My argument is that in blocks of 30 this table is of significance because it is the only table that excludes multiples of 3 and 5 (other than larger blocks of 30 that is).

    Yeah maybe but I think it’s because when using the sieve of Eratosthenes you only have to check multiples of primes upto sqrt(n), when n=30 you need to check upto 5.48, so 2, 3 and 5 (and by only writing down odd numbers you implicitly don’t include multiples of 2). It would then follow as 30 = 2*3*5 and those are the first three primes that you would get your pattern for any multiple of 30 (incidentally you can get a palindromic pattern by using 6 and taking out 3 and only writing odd numbers – albeit not a very interesting one, although I suspect many of the interesting results you claim would still hold). Maybe there is something interesting here but I don’t see it, having said that the maths I do isn’t number theory so what do I know?

    As a final note I really think you want to reconsider your comment about writing it solely for a layman. Mathematical notation has evolved over time to be clear (for other people who know what is happening) and reduce the chances of making errors in your reasoning. it isn’t simply used to be obtuse. Although wanting it to appeal to a common audience is admirable it isn’t worth it if only the laymen understands what you are doing. In addition as someone who is attatched to an intedisciplinary field (phylogenetics), although I’ve only been published once, you don’t just write a paper for publishing and say to hell with the audience, if you are submitting to a biology journal you play up the biology and down the math, and vice versa if you submit to a mathematics journal. You should really consider the possibility of having two versions, one for the layman and one using proper notation and reasoning particularly to ensure you haven’t made mistakes in your reasoning.

  • Thanks for your constructive comments. I did look at doing something similar to your suggestion about having two versions and tried to enlist the help of some friends. I have worked on submitting papers before when I was working in medical research many years ago and know that terminology is important. The hurdles proved to be too great in the end and I had a time constraint. Perhaps I might find someone who can help with this later on.

    After reading your comments I decided to look at the scatter plot of pairs of primes that form even integers and quickly realised that I had overlooked something because I hadn’t understood if before. I have posted a new paper about this – here is the link ;
    http://simplicityinstinct.weebly.com/uploads/4/7/3/3/4733019/goldbachconjecturebands.pdf

    It provides further support for the importance of the prime number channels but also creates some very interesting questions about what the minor bands actually mean.