Let me give you two analogies to start your understanding:
A) Integers are either composite or prime; think of their relationship rather like a block print and the resulting print: they both have patterns on them but it is the resulting print that we all look at and admire; we look at the block and can make no sense of it.
Humans have been looking at the prime “print” for a long time and made little sense of it; my answer is to look at the composites – it is quite eye-opening!
B) The prime pattern is an interference pattern between waves; you can find references to music and diagrams of waves but few appear to have taken if far. This is another line of attach worth examining but only after A)
In summary, what have I to report:
- The prime “pattern” expressed by prime numbers is in fact a composite of one to many individual patterns, each arising from a string of composites…..derived from each prime number – there is one relatively “simple” pattern for each prime number – OK not so simple and only the early few are short.
- The prime “pattern” is of a stepped nature so it starts simple (2,3….how more simple do you want it…..5….7?) and gets more complex with each step. The steps are at Pi2.
- Each “pattern” of composites that contribute to the overall “pattern” are of length Pi#.
- Each “pattern” of composites does not come into effect at Pi2. Each starts at Pi2 + a variable multiple of Pi.
- There is a drawbridge effect so that the overall pattern “progresses” and the distribution of the smaller primes are too simple to be included in the pattern for larger primes. This is one of the main reasons why virtually every rule for the primes we can define has an exception…..e.g. all primes are odd EXCEPT 2!