Following on from the previous post, here are a few thoughts about twin primes and frequencies.
I was picturing primes in terms of frequencies between 0 and 1. We can also reduce each set of twin primes to a single frequency, by looking at their product.
If we start by dividing the line into sixths, this gives us the product of 2 and 3 (which obviously can't overlap with any twin primes other than the pair 3/5, and nor can any shorter frequency reached by dividing by 2 or 3).
Now we know that neither of the frequencies for 1/5 and 1/7 will intersect the frequency for 1/6. We also know that the largest frequency which will intersect with both 1/5 and 1/7 is 1/35.
This frequency is symmetrical within the space, never intersects with the frequency 1/6 (or for 1/36).
To sieve for twin primes, we could go up through each pair of 6n+1/6n-1 numbers 11 and 13, 17 and 19, 23 and 25, and add the frequency for their product - 143, 323, 575 etc. When any part of this frequency coincides with an existing frequency, this is not a twin prime. For instance 115/575 coincides with 7/35 (both = 1/5), so we know 23 and 25 cannot be a pair of twin primes.
I like this idea in theory because it reduces the problem of twin primes to a symmetrical problem in which we are looking at a single number each time, the semi-prime product of two potential twin primes, rather than the overlapping patterns of two different patterns. We know there is an infinitude of semi-primes, though this need not mean that there is an infinitude of twin prime pairs.
You can also visualise the sieve I used on the previous post working for twin primes - that worked by dividing each number by the square of each of its factors. If we go through each 6n+1/6n-1 product, then we need to divide by the square of any two of its factors - so we get the series 1/35, 1/143, 1/323 - but for 575 we only get 575/25 = 1/23.
Each time we get a number that is smaller than the fractions/frequencies we have got before, we have reached a new pair of twin primes - because 575 only gives us 1/23 we know if can't be a twin prime pair product. Similarly 1295 (35x37) gives us 1295/37^2/49 = 25/37 and 49/37, neither of which are smaller than previous twin prime frequencies.
So there are things to like about this way of exploring twin primes - however I'm not sure it is actually much use in practise - the sieve outlined above is really laborious as it depends on factorisation, so it is only interesting as a mental representation. And I can't see any way of applying the frequency model that gives us anything we didn't already know about twin prime distribution.
The one thing that does interest me is that both the prime sieve in the previous post and the twin sieve in this post can be described in terms of a repetition of the same transform over and over. For primes you simply test the next integer by shrinking it into the space between 0 and 1. For twin primes you test the next semi-prime product of two n and n+2 numbers by shrinking into the space between 0 and 1. (I only bothered with the 6n+1 and 6n-1 numbers but a more basic algorithm would be to test every instance of n(n+2) where n is an integer and n=n+2.
I'm a bit boring about the fractal nature of prime number patterns but this brings out that fractal nature for me - as I said in the previous set this way of depicting the number line makes it look a lot more like the construction of a Cantor set.