We want to define the prime numbers as a logical sequence with a workable formula for the nth term and a means of establishing what n is when we have a given integer. As such we should then be able to distinguish prime numbers from composites.
Barriers to doing this:
i) whilst the general trend of the primes becoming less frequent with increasing n is well known the frequency does not decrease uniformly; the terms appear to pulse with both gaps and areas of higher frequency.
ii) the well known twin primes (but no triplets (see Counting the Twins)
iii) the well known gaps – with evidence of increasing gaps with increasing n
Primes have been described as random, unruly, baffling reflecting the frustration of the observer; there has also been commentary on harmony, music and other references to spirals, waves and the like. Just what is going on? Explainable or inexplicable?
In addition there are several well known observations about the primes:
Almost all rules/observations require an exception e.g.
- all primes are odd, except 2. So even the simplest description has an exception!
- all primes are separate, except 2 & 3.
Which leads onto a more general problem:
- rules for smaller primes may not apply to all (i.e. larger) primes
- rules that appear to apply require exceptions of smaller primes
As you will read in this blog once one understands the nature of the prime number problem and the inherent complexity, things become clearer, at least conceptually. Understanding that structure gives me some confidence to predict that there will never be a practical prime formula and since I can say why. However, there does seem to be a way of counting primes – but not every prime – and thus making some sense of things.
The startling thing is that the analysis is relatively simple…….