JSH: Why of prime gaps
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From: Penny Hassett on 18 Jul 2010 02:57 JSH wrote: > Recently an idea I had several years ago about prime numbers seemed to > me to be actually an axiom about prime numbers themselves, which was > rather exciting and I suggest that it is surprising that the world has > either yawned or is not aware of how significant it is, but it among > other things quite simply explains the why of prime gaps. In this > post I'll quickly show how that is the case by focusing on the twin > primes. > > An example of twin primes is: 11, 13 > > The gap between them is exactly 2, and one way of looking at "why" is > to note that if for any odd prime p less than sqrt(11), if (11+2) mod > p is 0, then the prime gap can't occur. May seem trivial but it is > the key to understanding the twin prime gap. > > For instance look at 13, where the next prime is 17. That's because > (13+2) mod 3 = 0. > > That's it. It's the only reason. Mathematically THERE IS NO OTHER. > > But also notice it means that 13 mod 3 = 1 is needed. So if 13 didn't > allow itself to have 1 as a residue modulo lesser primes then that > would have been a second twin primes case, but that is ludicrous. How > could the prime 13 decide that it doesn't like a particular residue > mod 3? > > And that's your clue. If you understand that one thing then you can > grasp the why of twin primes and then of arbitrary even prime gaps, as > for instance for that gap of 4 between 13 and 17, you needed (13+4) > mod p, where p is an odd prime less than sqrt(13) to not be 0, for, > once again 3. > > Mathematics doesn't need anything else to say a prime gap is there! > There is ONLY one way to get a prime gap, which is for > > (p_1 + g) mod p_2 > > to NOT be 0 for any odd primes less than sqrt(p_1). If it is, then > that gap does NOT occur. > > So the requirement is that p_1 NOT equal -g mod p_2, for all primes > p_2 less than sqrt(p_1). > > If no residues are excluded or preferred by the primes then there will > always be cases where those conditions are met, which trivially proves > the Twin Primes Conjecture. > > (Think carefully and now you can disprove Goldbach's Conjecture > trivially as well, but without a counterexample likely to ever be > directly seen which is unsatisfying I admit.) > > Another way of looking at it is, if you're looking at bigger and > bigger primes p_1, and you are getting all these residues modulo > primes less than sqrt(p_1), and there is no preference by the primes > then just at random at times you will have cases where -g is not a > residue modulo ANY of those primes which will give you a prime gap. > > (Primes may here define random for the real world.) > > So those go out to infinity. > > What's interesting about this issue may be that people rarely talk > about the why of prime gaps so it seems like a hard problem, > especially if you wrap the prime distribution into it which you can do > with a false correlation. That is, as the count of primes drops as > you get bigger numbers, necessarily the count of twin primes will drop > as well, but the prime distribution itself is irrelevant as to the > reason of the "why" of twin primes or other prime gaps. > > > James Harris James, a post from you with real mathematics in it is very welcome but the mathematics in this post is so badly explained that a newcomer couldn't understand it. I don't understand why you wrote it. As an example, when I look up your Prime Residue Axiom* I get ## Given differing primes p1 and p2, where p1 > p2, there is no preference for any particular residue of p2 for p1 mod p2 over any other. Now consider the following Hypothesis proposed by my cat Socratic Prime Residue Hypothesis ## For any sufficiently large prime number p, p+i is always composite if i is less than p^(1/p) but the probability of p+i being prime is independent of i if i is greater than p^(1/p). Does your Axiom disprove his Hypothesis? I think they can both be true unless you can show me otherwise. *There was a proposal for a JSH Wiki a while ago which I didn't support because I think it should be started by you. Blogs are all very well but going to the front page of yours didn't find the definition I wanted and I had to use a search engine. I've said before that I wish you would do something like that so that I could know which of your proposals you still support. For example, in the past you have said that a) the integers extended by sqrt(2) gives the rational numbers b) the rational numbers extended by pi gives the real numbers and I don't know your current thinking on these.
From: Penny Hassett on 18 Jul 2010 03:07 JSH wrote: > Recently an idea I had several years ago about prime numbers seemed to > me to be actually an axiom about prime numbers themselves, which was > rather exciting and I suggest that it is surprising that the world has > either yawned or is not aware of how significant it is, but it among > other things quite simply explains the why of prime gaps. In this > post I'll quickly show how that is the case by focusing on the twin > primes. > > An example of twin primes is: 11, 13 > > The gap between them is exactly 2, and one way of looking at "why" is > to note that if for any odd prime p less than sqrt(11), if (11+2) mod > p is 0, then the prime gap can't occur. May seem trivial but it is > the key to understanding the twin prime gap. > > For instance look at 13, where the next prime is 17. That's because > (13+2) mod 3 = 0. > > That's it. It's the only reason. Mathematically THERE IS NO OTHER. > > But also notice it means that 13 mod 3 = 1 is needed. So if 13 didn't > allow itself to have 1 as a residue modulo lesser primes then that > would have been a second twin primes case, but that is ludicrous. How > could the prime 13 decide that it doesn't like a particular residue > mod 3? > > And that's your clue. If you understand that one thing then you can > grasp the why of twin primes and then of arbitrary even prime gaps, as > for instance for that gap of 4 between 13 and 17, you needed (13+4) > mod p, where p is an odd prime less than sqrt(13) to not be 0, for, > once again 3. > > Mathematics doesn't need anything else to say a prime gap is there! > There is ONLY one way to get a prime gap, which is for > > (p_1 + g) mod p_2 > > to NOT be 0 for any odd primes less than sqrt(p_1). If it is, then > that gap does NOT occur. > > So the requirement is that p_1 NOT equal -g mod p_2, for all primes > p_2 less than sqrt(p_1). > > If no residues are excluded or preferred by the primes then there will > always be cases where those conditions are met, which trivially proves > the Twin Primes Conjecture. > > (Think carefully and now you can disprove Goldbach's Conjecture > trivially as well, but without a counterexample likely to ever be > directly seen which is unsatisfying I admit.) > > Another way of looking at it is, if you're looking at bigger and > bigger primes p_1, and you are getting all these residues modulo > primes less than sqrt(p_1), and there is no preference by the primes > then just at random at times you will have cases where -g is not a > residue modulo ANY of those primes which will give you a prime gap. > > (Primes may here define random for the real world.) > > So those go out to infinity. > > What's interesting about this issue may be that people rarely talk > about the why of prime gaps so it seems like a hard problem, > especially if you wrap the prime distribution into it which you can do > with a false correlation. That is, as the count of primes drops as > you get bigger numbers, necessarily the count of twin primes will drop > as well, but the prime distribution itself is irrelevant as to the > reason of the "why" of twin primes or other prime gaps. > > > James Harris James, a post from you with real mathematics in it is very welcome but the mathematics in this post is so badly explained that a newcomer couldn't understand it. I don't understand why you wrote it. As an example, when I look up your Prime Residue Axiom* I get ## Given differing primes p1 and p2, where p1 > p2, there is no preference for any particular residue of p2 for p1 mod p2 over any other. Now consider the following Hypothesis proposed by my cat Socratic Prime Residue Hypothesis ## For any sufficiently large prime number p, p+i is always composite if i is less than p^(1/p) but the probability of p+i being prime is independent of (i mod p) if i is greater than p^(1/p). Does your Axiom disprove his Hypothesis? I think they can both be true unless you can show me otherwise. *There was a proposal for a JSH Wiki a while ago which I didn't support because I think it should be started by you. Blogs are all very well but going to the front page of yours didn't find the definition I wanted and I had to use a search engine. I've said before that I wish you would do something like that so that I could know which of your proposals you still support. For example, in the past you have said that a) the integers extended by sqrt(2) gives the rational numbers b) the rational numbers extended by pi gives the real numbers and I don't know your current thinking on these.
From: Mark Murray on 18 Jul 2010 03:31 On 18/07/2010 08:07, Penny Hassett wrote: > *There was a proposal for a JSH Wiki a while ago which I didn't support > because I think it should be started by you. Blogs are all very well but > going to the front page of yours didn't find the definition I wanted and > I had to use a search engine. I've said before that I wish you would do > something like that so that I could know which of your proposals you > still support. For example, in the past you have said that > > a) the integers extended by sqrt(2) gives the rational numbers > b) the rational numbers extended by pi gives the real numbers > > and I don't know your current thinking on these. I doubt he does either. The closest you'll get is his Scribd pages at http://www.scribd.com/jstevh but this doesn't cover the two specific points above. As you have noted, James is very unclear in his writing, and is on record as not being able to follow some of his own work. He edits chaotically, so it is often not clear what he currently still believes during the course of a thread, and later as he revisits the subject. Having someone other than JSH put up a Wiki/Blog would allow a modicum of clarity into the collection by at least attempting to correct the above faults. It would also allow a separation of the mathematics from the hubris. M -- Mark "No Nickname" Murray Notable nebbish, extreme generalist.
From: Penny Hassett on 18 Jul 2010 06:08 Mark Murray wrote: > On 18/07/2010 08:07, Penny Hassett wrote: >> *There was a proposal for a JSH Wiki a while ago which I didn't support >> because I think it should be started by you. Blogs are all very well but >> going to the front page of yours didn't find the definition I wanted and >> I had to use a search engine. I've said before that I wish you would do >> something like that so that I could know which of your proposals you >> still support. For example, in the past you have said that >> >> a) the integers extended by sqrt(2) gives the rational numbers >> b) the rational numbers extended by pi gives the real numbers >> >> and I don't know your current thinking on these. > > I doubt he does either. > > The closest you'll get is his Scribd pages at > http://www.scribd.com/jstevh but this doesn't cover the two specific > points above. > > As you have noted, James is very unclear in his writing, and is on > record as not being able to follow some of his own work. He edits > chaotically, so it is often not clear what he currently still believes > during the course of a thread, and later as he revisits the subject. > > Having someone other than JSH put up a Wiki/Blog would allow a modicum > of clarity into the collection by at least attempting to correct the > above faults. It would also allow a separation of the mathematics from > the hubris. > > M Would you copy his work verbatim or would you try to interpret what he wrote? I don't know which faults you are trying to compensate for but I observe that he put up what he said was a provably NP solution to the traveling salesman problem but when I asked him for a proof he fobbed me off with a comment that it was obvious if one looked at the phase space, whatever that meant. Having a wiki isn't going to make his writing any clearer but will make it harder to rewrite history.
From: MichaelW on 18 Jul 2010 06:31
On Jul 18, 8:08 pm, Penny Hassett <Penny_Hass...(a)invalid.invalid> wrote: > Mark Murray wrote: > > On 18/07/2010 08:07, Penny Hassett wrote: > >> *There was a proposal for a JSH Wiki a while ago which I didn't support > >> because I think it should be started by you. Blogs are all very well but > >> going to the front page of yours didn't find the definition I wanted and > >> I had to use a search engine. I've said before that I wish you would do > >> something like that so that I could know which of your proposals you > >> still support. For example, in the past you have said that > > >> a) the integers extended by sqrt(2) gives the rational numbers > >> b) the rational numbers extended by pi gives the real numbers > > >> and I don't know your current thinking on these. > > > I doubt he does either. > > > The closest you'll get is his Scribd pages at > >http://www.scribd.com/jstevhbut this doesn't cover the two specific > > points above. > > > As you have noted, James is very unclear in his writing, and is on > > record as not being able to follow some of his own work. He edits > > chaotically, so it is often not clear what he currently still believes > > during the course of a thread, and later as he revisits the subject. > > > Having someone other than JSH put up a Wiki/Blog would allow a modicum > > of clarity into the collection by at least attempting to correct the > > above faults. It would also allow a separation of the mathematics from > > the hubris. > > > M > > Would you copy his work verbatim or would you try to interpret what he > wrote? I don't know which faults you are trying to compensate for but I > observe that he put up what he said was a provably NP solution to the > traveling salesman problem but when I asked him for a proof he fobbed me > off with a comment that it was obvious if one looked at the phase space, > whatever that meant. Having a wiki isn't going to make his writing any > clearer but will make it harder to rewrite history. The idea is not to copy his work (plenty of copies out there) but rather collect rebuttals and related information. This sort of thing is often only found on old Usenet threads. As you say it is a corrective to JSH's rewritting of history and dodging of rebuttals. Regards, Michael W. |