Monthly Archives: May 2013
Can you help me discover something?
I am placing these notes into the public domain with a view to identifying an appropriate and qualified collaborator(s). If you, yourself have skills and with formal mathematical or computational proofs perhaps we could collaborate. Otherwise if you have friends you think may be interested then please pass it on.
As an enthusiast for science and mathematics I quite enjoy employing otherwise unused brain processing time on interesting puzzles. Rather than choose brain teasers, I have chosen to play with puzzles that have “no answers – written at the back of the book”, in fact I have chosen to play with some of the unsolved or cumbersome proofs. Two of these where I feel I have had some success are, prime numbers and the proof of the four color theorem.
The truth is I have possibly spent thousands of hours trying to understand the prime numbers and somewhat less on the four colour theorem.
Here is a basic summary of my current understanding on these two puzzles. These notes assume a good understanding of these subjects.
If on reading this you believe you or someone else you know, has already reached this level of understanding, please let me know so I can review their work. If they pique you or a friend’s interest and can demonstrate skills with formal mathematical or computational proofs, perhaps we could collaborate.
The Four Color theorem:
As I understand the current situation there has been a successful proof developed, that first used a computer to test all the required combinations. This was considered controversial because the proof could not be demonstrated by any other means than a computer. Subsequently this proof was improved upon by demonstrating “a smaller set of combinations need be tested to prove the theory”.
My own work suggests a conceptual frame work, based on well know geometry, exists, that explains why the Four colour theorem is true, without the need to resort to a combinatorial proof, which simplifies this substantially, in fact almost to a triviality. Further, I have developed a process that may be deployed on any two dimensional map to colour it according to the requirements of the four colour theorem, that is without any two adjacent regions having the same colour. This process does not appear to be subject to any discontinuities or infinite regresses (for want of better words). This process appears also to demonstrate a proof for the four colour theorem by demonstrating how no combination of map can confuse the process.
The Prime numbers
There are a range of proofs and outstanding questions relating to the primes with many thinking of the prime numbers as a “very complex system”. This includes special cases such as “prime pairs”.
In my endeavour to understand the primes, I have developed a process to generate prime numbers in an extremely rapid manner, sorting primes from the non-primes at an increasingly fast rate. Interestingly I now understand the prime numbers as forming quite a simple pattern, repeated over and over but dependant at each step, on what went before.
I believe that the pattern and process in the prime generator is relatively simple once understood, and stands to support or deliver proofs to a number of questions about the primes. One example is, it clearly provides a proof for an infinite number of prime pairs, yet explains why we may not see them for a very long time when trawling through the large primes. The pattern also sheds light on the size of the prime gaps, including the maximum prime gaps found at a given distance into the numbers.
The process to generate the prime numbers I have developed, depends only on multiplication (that can be replaced by addition) and one or two simple transformations. The process needs minimal data to store the results and is re-start-able using the stored results. The process can generate a prime testing algorithm of 100% or diminishing accuracy.
There exists numerous glimpses of solutions to other problems and the opportunity to learn something about any given N without knowledge of the proceeding primes.
The Next steps ?
I have been unable to test these ideas using very large numbers due to a lack of skill and or familiarity with formal mathematical or computational proofs. However I believe the proofs lie within the samples I have used. It would however be wise to extend these tests further, in the hope of identifying flaws, it is quite easy to construct tests (or use some existing ones) that should quickly disprove the ideas – if they are in fact wrong.
Related Phrases and keywords
Process to colour 2D maps 4 color theorem, 4 Colour Theorem, four color Theorem, four colour Theorem
The basic Prime number pattern, infinitely small subset of an infinitely large set, prime number as a fundamental structure, prime pairs, prime number set