JSH: Why of prime gaps
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From: David Bernier on 18 Jul 2010 19:06 Penny Hassett wrote: > JSH wrote: >> >>> 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). >> >> Huh? I'd guess that p^{1/p} would be roughly 1? So why not just >> write 1? >> >> Simplify. See? >> >> Simplify your argument and ask again, please. >> >>> Does your Axiom disprove his Hypothesis? I think they can both be true >>> unless you can show me otherwise. > > You're right, I must have misunderstood what he said. Perhaps I should > have written ... > > > 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^(1/100) but the probability of p+i being prime is > independent of i if i is greater than p^(1/100). I don't know if your cat is aware of new results on small prime gaps from 2005. On the arXiv website, there's a pre-print by Goldston, Pintz and Yildirim announcing the unconditional result: liminf_{n -> oo} (p_{n+1} - p_n)/log(p_n) = 0. In words: << there are arbitrarily large primes that are �unusually close � together. >> At arXiv: < http://arxiv.org/abs/math/0508185 > For large p, p^(1/100) should be much larger than log(p), so perhaps the Socratic Prime Residue Hypothesis is too strong. David Bernier
From: MichaelW on 18 Jul 2010 20:22 On Jul 19, 9:06 am, David Bernier <david...(a)videotron.ca> wrote: > Penny Hassett wrote: > > JSH wrote: > > >>> 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). > > >> Huh? I'd guess that p^{1/p} would be roughly 1? So why not just > >> write 1? > > >> Simplify. See? > > >> Simplify your argument and ask again, please. > > >>> Does your Axiom disprove his Hypothesis? I think they can both be true > >>> unless you can show me otherwise. > > > You're right, I must have misunderstood what he said. Perhaps I should > > have written ... > > > 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^(1/100) but the probability of p+i being prime is > > independent of i if i is greater than p^(1/100). > > I don't know if your cat is aware of new results on small prime gaps from 2005. > On the arXiv website, there's a pre-print by Goldston, Pintz and Yildirim > announcing the unconditional result: > liminf_{n -> oo} (p_{n+1} - p_n)/log(p_n) = 0. > > In words: > << there are arbitrarily large primes that are unusually > close together. >> > > At arXiv: > <http://arxiv.org/abs/math/0508185> > > For large p, p^(1/100) should be much larger than log(p), so perhaps > the Socratic Prime Residue Hypothesis is too strong. > > David Bernier David, Thanks for the reference. I am trying to absorb the article now and find that it mentions the "level of distribution of primes". I don't understand the term and Google has not been my friend. Are you able to provide a link and/or explanation? Regards, Michael W.
From: Joshua Cranmer on 18 Jul 2010 20:48 On 07/18/2010 04:20 PM, JSH wrote: > So you think that if p is large enough that p mod 3 might decide that > it should NOT be 1 or -1, when say p mod 5 is not 0? It's not impossible. > If primes do not work together to block that eventuality then a twin > prime WILL occur whenever it is the case that for all odd primes less > than sqrt(p), it is NOT true that -2 is a residue modulo p. If they don't. You don't have any evidence other than "it doesn't seem like it should be so" (in other words, "I say so"). > Don't get it? I use humanizing the primes to point out what they > CANNOT do. That is a very interesting definition of "humanizing" then, given what you say below... > Primes are not human beings. "To humanize" generally means "to give or cause to have the fundamental properties of a human." This often has the connotation of ascribing human foibles to things which generally don't appear to have them. One such foible, that you love to refer to a lot, is collusion. So if you are ascribing human characteristics, such as the ability to collude, to an inanimate number, what is to say that two numbers cannot "collude"? -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
From: JSH on 18 Jul 2010 21:18 On Jul 18, 5:48 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > On 07/18/2010 04:20 PM, JSH wrote: > > > So you think that if p is large enough that p mod 3 might decide that > > it should NOT be 1 or -1, when say p mod 5 is not 0? > > It's not impossible. I should have had p mod 5 is not -2. So the issue is, if p is large enough, can it be the case that p mod 3 will never be -1, if p - 2 mod 5 is not 0. You assertion--assuming you were thinking of the correct value for p mod 5 versus what I put in error--is that it's not impossible. > > If primes do not work together to block that eventuality then a twin > > prime WILL occur whenever it is the case that for all odd primes less > > than sqrt(p), it is NOT true that -2 is a residue modulo p. > > If they don't. You don't have any evidence other than "it doesn't seem > like it should be so" (in other words, "I say so"). So you hypothesize chains of primes working together relative to each other with rules like if p mod 3 = 2, then p mod 5 must equal 3? That rule by itself would disprove the Twin Primes Conjecture if it were enforced. > > Don't get it? I use humanizing the primes to point out what they > > CANNOT do. > > That is a very interesting definition of "humanizing" then, given what > you say below... > > > Primes are not human beings. > > "To humanize" generally means "to give or cause to have the fundamental > properties of a human." This often has the connotation of ascribing > human foibles to things which generally don't appear to have them. One > such foible, that you love to refer to a lot, is collusion. So if you > are ascribing human characteristics, such as the ability to collude, to > an inanimate number, what is to say that two numbers cannot "collude"? Numbers don't collude. They're numbers. Primes are primes. 3 and 5 do not work together to block twin primes from existing because they like you and really want to help you with your argument. They aren't considerate of your feelings and anxious to do what's necessary to remove those ornery twin primes that will pop up if they just can't understand your need to feel right. 3 and 5 cannot feel your pain. James Harris
From: David Bernier on 18 Jul 2010 23:22
MichaelW wrote: > On Jul 19, 9:06 am, David Bernier<david...(a)videotron.ca> wrote: >> Penny Hassett wrote: >>> JSH wrote: >> >>>>> 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). >> >>>> Huh? I'd guess that p^{1/p} would be roughly 1? So why not just >>>> write 1? >> >>>> Simplify. See? >> >>>> Simplify your argument and ask again, please. >> >>>>> Does your Axiom disprove his Hypothesis? I think they can both be true >>>>> unless you can show me otherwise. >> >>> You're right, I must have misunderstood what he said. Perhaps I should >>> have written ... >> >>> 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^(1/100) but the probability of p+i being prime is >>> independent of i if i is greater than p^(1/100). >> >> I don't know if your cat is aware of new results on small prime gaps from 2005. >> On the arXiv website, there's a pre-print by Goldston, Pintz and Yildirim >> announcing the unconditional result: >> liminf_{n -> oo} (p_{n+1} - p_n)/log(p_n) = 0. >> >> In words: >> << there are arbitrarily large primes that are �unusually >> close � together.>> >> >> At arXiv: >> <http://arxiv.org/abs/math/0508185> >> >> For large p, p^(1/100) should be much larger than log(p), so perhaps >> the Socratic Prime Residue Hypothesis is too strong. >> >> David Bernier > > David, > > Thanks for the reference. I am trying to absorb the article now and > find that it mentions the "level of distribution of primes". I don't > understand the term and Google has not been my friend. Are you able to > provide a link and/or explanation? For 0< theta < 1, the primes have level of distribution theta if their distribution in various congruence classes is relatively un-skewed. For a number q >= 1, there are phi(q) congruence classes modulo q that are susceptible to containing infinitely many primes. If q = 10, phi(10) = 4 because there are four numbers from 1 to 10 co-prime with 10: 1, 3, 7 and 9. Bombieri proved that theta=1/2 is true in the inequality (1.3) of the Goldston, Pintz and Yildirim paper ("level theta" is introduced there by: "We say that [...] level theta [...]", so it may be uncommon usage). Probably a good place to start is to read Wikipedia's article on Dirichlet's Theorem: < http://en.wikipedia.org/wiki/Dirichlet's_theorem_on_primes_in_arithmetic_progressions> I think Dirichlet only proved that all phi(q) admissible congruence classes contain infinitely many primes. For results on the distribution, even for the trivial progression 1, 2, 3, .... (q = 1), the Prime Number Theorem was only proved in 1896. The article on PNT says that de la Vall�e Poussin also proved PNT for primes in arithmetic progressions: < http://en.wikipedia.org/wiki/Prime_number_theorem >. The Elliott-Halberstam Conjecture is about the best that can be hoped for: any theta < 1 works. The article on the Elliott-Halberstam Conjecture shows how maximum discrepancies from the average are added up and compared to a reference number which grows with x. < http://en.wikipedia.org/wiki/Elliott�Halberstam_conjecture > . According to Goldston, Pintz and Yildirim, theta = 1/2 in their (1.3) is the statement of Bombieri-Vinogradov Theorem (however in (1.1) they give the weight log(n) to the prime n, whereas the pi(x) prime-counting function gives a weight 1 to every prime). David Bernier |