First, P2s are more common than primes, by a factor of log log n. But more importantly adding in P2s allows the use of sieve theory which has traditionally suffered from the parity problem. There are some new techniques which break the parity barrier but they're not easy to use. So I agree with this unnamed mathematician. Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Sep 7, 2012 at 3:47 PM, Warren Smith <warren.wds@gmail.com> wrote:
A "P2" is the product of exactly 2 primes. A "P3" ditto for 3 primes etc.
The most that is rigorously known about upper-bounding prime gaps is, there is always a prime between X and X+X^(21/40+epsilon) if X is sufficiently large, for any fixed epsilon>0.
A well-known mathematician suggested to me that perhaps a much stronger upper bound would be possible (or was already known), e.g. bringing the exponent 21/40 way down, likely below 1/2, 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; adding P2's to the primes should have almost no effect on typical gap sizes asymptotically. And for that matter adding P3s, P4s, .. etc up to any fixed number of allowed factors also should have almost no effect.
A random large number N has lnlnN prime factors on average, with standard deviation sqrt(lnlnN), see http://en.wikipedia.org/wiki/Erdos-Kac_theorem .
Comments from math-fun number theorists?
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