From: Gareth McCaughan <gareth.mccaughan@pobox.com>
On Friday 07 September 2012 20:47:08 Warren Smith wrote:
A "P2" is the product of exactly 2 primes. ...
The most that is rigorously known about upper-bounding prime gaps is, ... A well-known mathematician suggested to me that perhaps a much stronger upper bound would be possible (or was already known), [...] if we considered gaps between, not "primes," but rather "primes U P2's."
I counter-argued that since P2s are much rarer than primes, that seemed silly; [...]
Aside from what's already been pointed out, namely that P2s are commoner than primes rather than rarer, note the difference between (1) what the biggest gaps *are*, (2) what typical gaps are, and (3) the best bound we can prove on either.
I'm glad someone raised these points, as there's a huge disconnect between what we suspect to be true (that can be supported using believable heuristics) and what we can prove to be true. Life would be more satisfying once ERH had its i's crossed and its t's dotted, but even then the provable bounds still look dreadfully slack at omega(sqrt(p)), assuming Cramer's still the tightest ERH-conditional bound.
For instance: the numbers floor(n (log n)^2) are rarer than the primes "by about a factor log n" but the gap between two of these of size ~n is only on the order of (log n)^2, much smaller than the biggest gaps between consecutive primes; and if we take these numbers and perturb them in a suitable (well controlled) way we can make the biggest gaps between them *provably* about as big as the biggest gaps between primes are conjectured to be. The fact that these numbers are rarer than the primes, and therefore have bigger gaps on average, is neither here nor there.
Clearly your original set is the corner case with specifically chosen gaps and density, but of course it displays none of the traits of randomness that the primes have. Stirring in "suitable" perturbations still makes it a highly contrived set with all th properties you desire "just so". So I don't believe it helps give any insight into the primes at all, it's just an example of what the primes can never be like. Instead. What happens if we take the lucky (or ludic) numbers - what's known about their gaps? Phil