JSH: Why of prime gaps
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From: David Bernier on 22 Jul 2010 05:07 JSH wrote: > On Jul 17, 10:32 pm, David Bernier<david...(a)videotron.ca> wrote: >> JSH wrote: >> >> [...] >> >> >> >> >> >>> 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). >> >> [...] >> >> Sorry. I have a proof secured in a safe that, for some large >> constant C> 0 and for the case g = 2: >> >> "For every prime p_1> C, there exists a prime p_2 with >> p_1 == -2 (mod p_2) and with p_2< sqrt(p_1)." > > An assertion of prime preference--it's wrapped up in your actual > statement! Infinity is far. Primes are infinite in number. How many primes have you seen? You make a blanket assertion of prime non-discrimination, and expect us to believe. We are not in your Church. We are the skeptics. If p is a prime p>2, then (2^p -2) is divisible EXACTLY by p ... So why does 2^p prefer 2 as a residue mod p when p is an odd prime??? David Bernier > So your assertion is that at some critical level which you call C, the > primes above that level so dislike -2 as a residue that they avoid it > COMPLETELY and PERFECTLY out to infinity. > > I humanize the primes so that the choice aspect of your assertion is > clear. > > Why would THOSE primes decide that they are unlike their brethren at > lesser values and no longer can stand -2 as a residue? > > Did they tell you in any messages from that great level? Possibly a > spirit transmission to you in a dream? > > > James Harris >
From: Ostap Bender on 22 Jul 2010 05:33 On Jul 18, 8:15 am, JSH <jst...(a)gmail.com> wrote: > On Jul 17, 10:32 pm, David Bernier <david...(a)videotron.ca> wrote: > > > > > JSH wrote: > > > [...] > > > > 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). > > > [...] > > > Sorry. I have a proof secured in a safe that, for some large > > constant C > 0 and for the case g = 2: > > > "For every prime p_1 > C, there exists a prime p_2 with > > p_1 == -2 (mod p_2) and with p_2 < sqrt(p_1)." > > An assertion of prime preference--it's wrapped up in your actual > statement! > > So your assertion is that at some critical level which you call C, the > primes above that level so dislike -2 as a residue that they avoid it > COMPLETELY and PERFECTLY out to infinity. Why not? After all, all primes larger than p so dislike '0 mod p' as a residue that they avoid it COMPLETELY and PERFECTLY out to infinity. We all (maybe except for you) know the reason why they do so. So, why can't there be a reason why primes would tend to avoid other residues as well? > I humanize the primes so that the choice aspect of your assertion is > clear. > > Why would THOSE primes decide that they are unlike their brethren at > lesser values and no longer can stand -2 as a residue? Why would THOSE primes decide that they are unlike their brethren at > lesser values and no longer can stand 0 as a residue? > Did they tell you in any messages from that great level? Possibly a > spirit transmission to you in a dream? No, you are the only one (outside Bellview) who receives such transmissions.
From: Ostap Bender on 22 Jul 2010 05:39 On Jul 18, 1:20 pm, JSH <jst...(a)gmail.com> wrote: > 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? So you think that if p is large enough that p mod 3 might decide that it should NOT be 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. Or you can collude with Iran to manufacture nuclear bombs. However, both events are highly improbable. > 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. Can primes collude (work together) to avoid residue 0 mod 5? > Prime numbers are NOT HUMAN. So, there must be infinitely many primes which are 0 mod 5? Are you sure?
From: Ostap Bender on 22 Jul 2010 06:42 On Jul 18, 1:20 pm, JSH <jst...(a)gmail.com> wrote: > 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. Let us assume that asymptotically, all residues 1,2,3...p-1 do indeed occur with equal frequency among primes (which is most likely a well- known result in number theory anyway). How does that prove (and I mean RIGOROUSLY prove) that the number of twin primes is infinite? > > 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: JSH on 22 Jul 2010 10:25
On Jul 22, 2:07 am, David Bernier <david...(a)videotron.ca> wrote: > JSH wrote: > > On Jul 17, 10:32 pm, David Bernier<david...(a)videotron.ca> wrote: > >> JSH wrote: > > >> [...] > > >>> 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). > > >> [...] > > >> Sorry. I have a proof secured in a safe that, for some large > >> constant C> 0 and for the case g = 2: > > >> "For every prime p_1> C, there exists a prime p_2 with > >> p_1 == -2 (mod p_2) and with p_2< sqrt(p_1)." > > > An assertion of prime preference--it's wrapped up in your actual > > statement! > > Infinity is far. Primes are infinite in number. How many > primes have you seen? You make a blanket assertion of > prime non-discrimination, and expect us to believe. No, I also talk consequences of the OPPOSITE of non-preference. For instance, if for primes p greater than some arbitrary high limit p mod 7 tends to equal 2, then the residues of composites with those primes as factors will tend be 2^j, where j is a natural number. > We are not in your Church. We are the skeptics. I don't have a church. Then act like a skeptic. It's not just a word "skeptic". It's also a behavior. I'm challenging the status quo and an elevated position given to things like the twin primes conjecture and THAT is your dogma--this notion that twin primes aren't completely explained and the twin primes conjecture is trivially true. > If p is a prime p>2, then (2^p -2) is divisible > EXACTLY by p ... So why does 2^p prefer 2 > as a residue mod p when p is an odd prime??? There is number theory which will answer that question, but FLT is a non sequitur here. p mod 3 LOOK at it, you want it to have a structure and choice that mathematically has no reason to exist, and why? Because a little while after Gauss died some men got hot and bothered about prime gaps of 2, and made a simple error of missing the obvious. With over a hundred years of history you are in a "Church" created around that error which is just one of their mistakes as they made an even bigger one with the ring of algebraic integers. If you were reading about Ptolemy versus Kepler you'd chuckle at people like you who fought, fought, fought to hold on to those damn spheres. But in your position you are just like them, while years from now, students of mathematics will read about you with disbelief! How could you hold on to such an obvious error? > > David Bernier History is easier when you read about it than when you live it David. James Harris |