John Grieve: Twin Primes appear to be Primary    
 Twin Primes appear to be Primary1 comment
13 Oct 2010 @ 08:13, by John Grieve

Throughout the history of the theory of numbers, primes have been studied with an emphasis on the stand-alone prime numbers such as 23 and 37. That is, the individual isolated prime number, of which it is easy to prove that there are an infinite number. The other sort of prime number, twin primes such as 17,19 are similarly thought to be infinite but this is just a conjecture and has not been proven yet.

A few years ago I put forward a number of my own conjectures regarding the twin primes. I speculated about what I called "Points of Symmetry" whereby the twin primes were connected by a sort of symmetry [link]

It seems to me that instead of looking at all prime numbers as being chaotic, almost random and without much order, we should build on what is there. The fact that twin primes exist and there seem to be an infinite number of them suggests a high level of organization. I would like to point to four things specifically, even at the risk of repeating myself.

(1) The Goldbach Conjecture, if true,implies that every prime number is equidistant from two other prime numbers.

(2) I have conjectured on the basis of this fact, and some supporting evidence, that every pair of twin primes is equidistant from two other pairs of twin primes.

(3) I now add a further conjecture which seems to be supported by the facts: every stand-alone prime is equidistant from a twin-prime and another stand-alone prime.

(4)Finally, if you look at magic squares, particularly odd ones such a 3x3, 5x5, 7x7 etc. it is noticeable that there is often a striking symmetry in the lay-out of the twin primes.


In conclusion,I would say that in trying to understand prime numbers it may be time that we studied the obvious patterns that are there, particularly those presented by the twin primes and their points of symmetry.


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