Prime Number distribution has been a source of fascination to both professional and amateur mathematicians for many years. The distribution is easy to explain, and conceptually easy to understand, once explained, but almost impossible to see if one looks just at the prime numbers themselves.
I believe that considering prime numbers as fundamental may have stood in the way of understand prime number distribution as their distribution is the result of composite number patterns, and the interference of these simple patterns that leads to the relative complexity of patterns in the primes.
The picture above of the two rows of trees in an avenue is not by chance. If the row of trees at one prime spacing combines with the row of trees at a different prime spacing one gets a combined pattern. Primes are just the left overs after all composite patterns have been taken into account.
Imagine two waves – one of wavelength 2 and one of wavelength 3; consider their interference pattern after 9 (for <9 see later). A familiar pattern emerges: at intervals of 6 three integers are multiples of 2 and two of 3; the resultant left overs (or potential primes) occur in pairs: thus 11,13 followed by 17,19 etc. The occurrence of pairs of primes are one of the noteworthy features of the prime series, and they occur simply as a result of first two primes. As factors of higher primes kick in (see later) the numbers of pairs declines. The question are their more potential pairs than there are multiples of higher primes (in which case pairs of primes continue indefinitely) or are there more multiples of higher primes than potential pairs (in which case pairs of primes disappear, but at what point?)
So what are the steps in understanding? Firstly, whilst primes may be the building blocks for integers, their pattern is determined by the composites. Prime distribution is determined not by one simple pattern, but a changing pattern that becomes more complex as integer number increases and in an infinite way. Secondly, the overall pattern is a composite of what are relatively simple patterns, each produced by a single prime factor. Thirdly, the composite patterns come into play at different starting points.
That’s about it. Once you have these concepts and the details (see below), prime number distribution is easy to explain……however, once you realise that the “simple” patterns become very long, very quickly you will see why visualisation of any pattern is simply beyond the capacity of the simple human brain.
You can also explain why repeatable patterns do occur in the primes, and where they come from!
Thus the overall pattern between 4 and 9 is not the same as it is between 9 and 25.
Also smaller prime factors influence the arrangement of the pattern of larger prime numbers.
Thus by integer 25 the component pattern that joins the overall prime pattern repeats itself at a length of 30 (2x3x5); however, the repeating patern only starts at 35 (not 25), whilst at integer 49 another component joins with a repetition stretching to 210 (2x3x5x7) but the repeating pattern only starts at 77 (=49+4*7). By the time you get to 121 you’re looking at a further pattern being added with a repeating pattern of length 2,310.
I hope these short paragraphs give you a little more insight into how prime numbers are distributed. Feel free to rate any pages you find useful, or comment.