JSH: Why of prime gaps
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From: Mark Murray on 18 Jul 2010 15:17 On 18/07/2010 19:16, Joshua Cranmer wrote: > Numbers merely have properties prescribed to them by rigid formulae. > Anthropomorphizing them as to deciding whether or not they can choose to > have a value is pointless: their constructions dictate that they must > have a certain property, as explained by a formula, or not. Anthropomorphising numbers just irritates them. M -- Mark "No Nickname" Murray Notable nebbish, extreme generalist.
From: Nell Fenwick on 18 Jul 2010 15:29 "Mark Murray" <w.h.oami(a)example.com> wrote in message news:4c435325$0$12166$fa0fcedb(a)news.zen.co.uk... > On 18/07/2010 19:16, Joshua Cranmer wrote: >> Numbers merely have properties prescribed to them by rigid formulae. >> Anthropomorphizing them as to deciding whether or not they can choose to >> have a value is pointless: their constructions dictate that they must >> have a certain property, as explained by a formula, or not. > > Anthropomorphising numbers just irritates them. > > M > -- > Mark "No Nickname" Murray > Notable nebbish, extreme generalist. don't say that too loud, I have had many a number POed at me.
From: JSH on 18 Jul 2010 16:08 On Jul 17, 11:57 pm, Penny Hassett <Penny_Hass...(a)invalid.invalid> wrote: > 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. I'm simplifying some concepts around prime residues so that people can better understand my prime residue axiom, as clearly based on some recent replies to me, some of its consequences--like allowing trivial proof of the Twin Primes Conjecture and disproof of Goldbach's Conjecture--are missed on at least some people. Simplifying is NOT necessarily easy but I think it's a worthwhile effort as it helps me check my own understanding. And OFTEN when simplifying one can actually end up tossing a failed argument as you finally realize that it does not work!!! > 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. My original post is an attempt at simplification. Obfuscation is not something I think is a social good. > *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 No. sqrt(2) is an object. Google: object ring > b) the rational numbers extended by pi gives the real numbers Yes. But the rational numbers gives the reals as well. That's by my set theory which I pulled off the web. If you introduce 1/2 into the set of integers you get reals. Classical mathematics disagrees. > and I don't know your current thinking on these. Now you do. James Harris
From: JSH on 18 Jul 2010 16:20 On Jul 18, 11:16 am, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > On 07/18/2010 11:15 AM, JSH wrote: > > > Why would THOSE primes decide that they are unlike their brethren at > > lesser values and no longer can stand -2 as a residue? > > Have you ever heard of "Strong's Law of Small Numbers"? It is a law > which notes that there exist many things which appear to be patterns for > a while... until they completely destroy it. A famous example is the > number of regions you can divide a circle by n chords. It goes 1, 2, 4, > 8, 16... 31. > > Why should 5 have suddenly decided it didn't want to divide a circle > into 32 regions? You might argue that the sixth number of a pattern is > too small to compare to primes, but then what is that magic number? 10? > 100? 1 million? 1 googolplex? Grahm's number? p mod 3 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? But if p mod 3 doesn't care what you think, and neither does p mod 5 and the primes are not colluding together--a prime cabal!!!--then eventually there will occur this thing where p mod each prime less than sqrt(p) will NOT equal -2, and then guess what? You'll have a twin prime. That is absolute. It's ALL that's necessary. 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. > Numbers merely have properties prescribed to them by rigid formulae. > Anthropomorphizing them as to deciding whether or not they can choose to > have a value is pointless: their constructions dictate that they must > have a certain property, as explained by a formula, or not. Correct. My point exactly. Don't get it? I use humanizing the primes to point out what they CANNOT do. Primes are not human beings. You on the other hand are a human being, correct? So YOU can collude with other posters, say, to claim that my research is not valid and you can work together with them to push this notion regardless of the evidence. But prime numbers are NOT human. They do not work together to block twin primes at some arbitrary level. They just don't do it at all. Prime numbers are NOT HUMAN. Re-read my post carefully. You just agreed with me in principle. James Harris
From: Penny Hassett on 18 Jul 2010 19:07
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). Does your Axiom disprove his Hypothesis? I think they can both be true unless you can show me otherwise. |