Have been thinking about the twin primes lately, dabbling with the Twin Prime Conjecture, which is that there is an infinite number of twin primes, where a twin prime is two primes separated only by 2, with a single even natural number between.
In a similar spirit of numerical experimentation as what led to the previous findings and draft paper, I started looking at patterns in the occurrence of twin primes.
I thought there might be some way to show that there must exist, within any given (nth) primorial P_n#, at least one unique twin prime pair, or in other words, that between P_(n-1)# and P_n# there exists at least one twin prime pair. This would be sufficient to prove the Twin Prime Conjecture, since for every prime, there exists a corresponding primorial, and there are infinitely many primes.
Intuitively, there MUST be a least one twin prime pair between every two sequential primorials, and in fact there are many of them, and the number grows as n grows, by observation. But how does one PROVE the existence of at least one twin prime as claimed?
Given that it’s hard to predict where primes will occur, it’s doubly difficult to predict where twin primes will occur. I thought it might be easier to work with a smaller space, within which the inputs are less numerous, and therefore easier to manage. If instead of looking for twin primes as described above, what if we narrow our focus to the gap between adjacent squares of primes. If one could show that between P_(n-1)^2 and P_n^2 there MUST exist at least one twin prime pair, then again the conjecture would be proven.
While numerical experimentation SUGGESTS this is true, as shown below, I’m still not sure it would be easy to PROVE. In any event, it is possible to methodically count the number of twin prime pairs between successive prime squares, P_n^2, and plot them as follows (for prime squares up to 63001, or 251^2):
By inspection of the graph, there appears to be minimum and maximum trends, corresponding to about 1/7 (lower bound) and 4/3 (upper bound). However, who knows whether, as the primes extend out to infinity, this chaotic pattern swings down to zero somewhere, which would indicate zero twin primes in a gap between prime squares. One zero alone would not disprove the conjecture, as it would be necessary for ALL gaps beyond some point to contain zero prime pairs for the number of twin pairs to be bounded. But still, a trend on a graph, while perhaps intuitively compelling, is not proof.
The reason I considered looking at the gaps between prime squares is because, below P_n^2, the only factors involved in eliminating (i.e. sieving) lower primes are the primes up to P_n. So theoretically, you could examine the superposition of the harmonics associated with all the primes to P_n, and focus your attention only in the region between P_(n-1)^2 and P_n^2 to look for the necessity of existence of a prime pair.
I may look at this a little more closely at a later time, but for now I’d like to turn my attention back to the space between successive primorials, which is more open, with seemingly more potential to prove the existence of a single prime pair. By playing with numbers, I’ve found (and it’s obvious in retrospect from looking back at my older graphs of possible primes) that the possible locations of prime pairs are fixed within each primorial, and are symmetric within the primorial, as was the case with the primes themselves.
This is not to say that the distribution of prime pairs is symmetric in any way… just the POSSIBLE prime pairs, in the same sense as I’ve used the term, “possible prime,” are symmetric within any primorial.
The following graph shows a large number of “twin prime centres” taken as (mod 30), where the “twin prime centre” is the even number between the twin primes. You can see that, within every block of 30 natural numbers, there are only three places where a twin prime can occur: right at the multiple of the primorial (30) itself, or at 12 or 18 positions from the multiple of the primorial:
This result makes perfect sense i we recall the distribution of possible primes from before, and note that a twin prime can only occur where there were adjacent (i.e. separated by 2) possible prime positions within a given primorial. For reference, we can look at the distribution of possible primes and possible prime pairs within repetitions of 30:
Extending the inquiry, we can look at possible twin primes within repetitions of 210 (with possible primes also shown for reference):
and the same for 2310 (this time without possible primes shown):
Now I suppose this doesn’t really mean much. It seems like there are LOTS of twin prime pairs associated with each primorial, so it should be hard to imagine that surely there MUST be at least one. But wishing and imagining are not the same as proving, so at best this is fairly good numerical evidence, but far from being proof. If we look far enough, I suppose it’s possible the twin primes start to thin out.
As a final quick look at the numerical evidence, the next two graphs show the distribution of twin primes between successive primorials. The first graph plots number of twin prime pairs against the ordinal, n, of the prime, or the prime itself. You can see the number of twin primes grows very rapidly with n.
And this graph shows number of twin prime pairs against the value of the higher primorial, plotted in log-log scale:
So, nothing proven, but I think there are some interesting patterns.
I’ll close with one more observation. All twin primes straddle an even number, since primes other than 2 are all odd. It is quite straightforward to show that all prime pair centres are not only even, but must be multiples of 6 (the proof is trivial). From there you can easily generalize that all twin prime centres must also be multiples of a primorial, since all primorials greater than 6 are multiples of 6. But by inspection of these graphs, evidently not all multiples of primorials must necessarily be possible prime pair centres, and many are removed at each successive primorial, as was the case with the primes.