[math-fun] gaps between primes, P2s, P3s, etc
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?
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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Yes, obviously semiprimes are more common than primes. For every prime p, we have the semiprimes {2p, 3p, 5p, 7p} (and others). This means that if the primes locally have a density of d, the semiprimes have a density > (1/2 + 1/3 + 1/5 + 1/7)d > d. There are not merely linearly more semiprimes than primes, either, since the prime harmonic series 1/2 + 1/3 + 1/5 + 1/7 + ... diverges. The same argument shows that there are asymptotically more P3s than semiprimes, and more P4s than P3s, and so on ad infinitum. Sincerely, Adam P. Goucher http://cp4space.wordpress.com/ ----- Original Message ----- From: "Charles Greathouse" <charles.greathouse@case.edu> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Friday, September 07, 2012 9:04 PM Subject: Re: [math-fun] gaps between primes, P2s, P3s, etc
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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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. 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. -- g
participants (4)
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Adam P. Goucher -
Charles Greathouse -
Gareth McCaughan -
Warren Smith