Goldbach's Conjecture Scatter Plot Deciphered

By Guest Author 02/08/2010


By Phil Jackson

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.

Banded Patterns of Pairs of Prime Numbers versus Positive Integers up to 1e+06
Banded Patterns of Pairs of Prime Numbers versus Positive Integers up to 1e+06

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.