JSH: Prime gaps examples
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From: Joshua Cranmer on 21 Jul 2010 21:41 On 07/21/2010 08:24 PM, JSH wrote: > Is there a preference involved? > > No. So you have the prime residue axiom which says the primes HAVE NO > PREFERENCE. > > But logically you know they can't as composites are products of > primes. If say, primes modulo 3 had a preference for, say, 1, then > guess what? Composites would reflect the SAME PREFERENCE. They have > NO CHOICE. They are PRODUCTS OF THE PRIMES. Let's think critically about your argument. Your argument, as I understand it, boils down to "the residues of primes must be uniformly distributed because otherwise the residues of composites would not be uniformly distributed." Now, consider two cases: 1. k = 0 mod 3. Almost no primes have a residue of 0 mod 3, so they have a relative preference for 1 and 2 mod 3. By your argument, that means that composites should also reflect this relative preference of not being 0 mod 3. Yet we know that 1/3 of all numbers are 0 mod 3--so clearly, there is another factor at work that is mitigating this preference. 2. Consider k mod 6. If we exclude multiples of 6--which we know to never be prime--we should find that 1/5 have a particular residue. Trivially, however, a prime must be 1 or 5 mod 6. We also know that of the other four possible values, 0 and 4 mod 6 can never be prime, and there is only 1 prime each for 2 and 3 mod 6. Yet, once again, numbers are equally distributed among the residues. Again, something must be mitigating this preference. Undoubtedly, these are special cases. But it does show that it is possible to mitigate a "preference". There is no a priori reason to believe that an uneven distribution in residues cannot exist: in fact, one already pretty blatantly exists without bringing the whole of mathematics down. Perhaps it could be shown that this mechanism is limited only to these special cases; such a step would probably be a major stepping stone to the twin primes conjecture (it's not sufficient though--you would also have to show that these probabilities are independent). Shouting that something is true does not make it true, nor does its seeming obviousness. The human thought process likes patterns and order and will synthesize them where they do not really exist. > (If you buy a book on twin primes--well, maybe you should consider > that you're not exactly smart. Or maybe you could just give your > money away to people on the street. It might do more that way.) In one fashion, intelligence is recognizing that what appears to be true is not. The world looks to be flat from my eyes, but I know it to not be so. It appears to require constant force to keep an object moving at the same speed--but it really requires no force at all. -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
From: JSH on 21 Jul 2010 23:49 On Jul 21, 6:41 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > On 07/21/2010 08:24 PM, JSH wrote: > > > Is there a preference involved? > > > No. So you have the prime residue axiom which says the primes HAVE NO > > PREFERENCE. > > > But logically you know they can't as composites are products of > > primes. If say, primes modulo 3 had a preference for, say, 1, then > > guess what? Composites would reflect the SAME PREFERENCE. They have > > NO CHOICE. They are PRODUCTS OF THE PRIMES. > > Let's think critically about your argument. Your argument, as I > understand it, boils down to "the residues of primes must be uniformly > distributed because otherwise the residues of composites would not be > uniformly distributed." Now, consider two cases: > > 1. k = 0 mod 3. Almost no primes have a residue of 0 mod 3, so they have Only one does. 3. > a relative preference for 1 and 2 mod 3. By your argument, that means > that composites should also reflect this relative preference of not > being 0 mod 3. Yet we know that 1/3 of all numbers are 0 mod 3--so > clearly, there is another factor at work that is mitigating this preference. Uh, yeah, they're divisible by 3. > 2. Consider k mod 6. If we exclude multiples of 6--which we know to <deleted> Yet 6 is NOT prime. One thing that often fascinates me about arguments about mathematics is, when it occurs to me that a poster has no interest whatsoever in what the correct mathematics actually is!!! And I asked in a thread, why bother doing math research? Didn't get much in answer. Not surprised. These people don't care. The answer for such people appears to be that they don't bother. But they like to argue on Usenet for phantom approval, with occasional words of approval from other people like them who also don't care about what is correct. So what happens are endless nonsense arguments because these people don't give a damn what the mathematics actually says. But one day they'll be dead. Their arguments gone. The emotions that drive this silly behavior not even a memory. As their bodies turn to dust, who will care what arguments they made about prime numbers, and what thoughts drifted through their minds.... Mathematics is a hard discipline because it ultimately belongs to those who DO care about what is correct. Who don't give a damn what people think and don't give a damn what people remember. It can only belong to them in that only they can find the truth of it. For others there's just what people say, or even worse--what they believe. The searchers just want to know the truth. One day maybe someone will invent some juice to inject into the veins of people who just want to feel important. And they can sigh with ecstasy in the belief of their math knowledge, and be content--and the Usenet newsgroups can be quieter. But there will be some who will never take such drugs to their veins. But they would not live in fantasy now either. Who cares? So some math people are fakes or deluded and are way wrong but a bunch of other people believed in them. When they're all dead, who will remember? James Harris
From: Tim Little on 22 Jul 2010 00:37 On 2010-07-22, JSH <jstevh(a)gmail.com> wrote: > One thing that often fascinates me about arguments about mathematics > is, when it occurs to me that a poster has no interest whatsoever in > what the correct mathematics actually is!!! Gee, how ironic. I noticed you snipped out the actually relevant mathematics to embark on your rant. Here it is again: When you count the primes less than K with residue 1 vs residue 2 mod 3, the 2s are in front for almost all values of K. Why is that if your claim of "no preference" was correct? What *in mathematical language* are you actually claiming? [snipped 44 lines from JSH with zero math content] - Tim
From: Ostap Bender on 22 Jul 2010 05:08 On Jul 21, 8:49 pm, JSH <jst...(a)gmail.com> wrote: > On Jul 21, 6:41 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > > > > > On 07/21/2010 08:24 PM, JSH wrote: > > > > Is there a preference involved? > > > > No. So you have the prime residue axiom which says the primes HAVE NO > > > PREFERENCE. > > > > But logically you know they can't as composites are products of > > > primes. If say, primes modulo 3 had a preference for, say, 1, then > > > guess what? Composites would reflect the SAME PREFERENCE. They have > > > NO CHOICE. They are PRODUCTS OF THE PRIMES. > > > Let's think critically about your argument. Your argument, as I > > understand it, boils down to "the residues of primes must be uniformly > > distributed because otherwise the residues of composites would not be > > uniformly distributed." Now, consider two cases: > > > 1. k = 0 mod 3. Almost no primes have a residue of 0 mod 3, so they have > > Only one does. 3. Really? Amazing! Is this your conjecture, or can you actually prove it! Even Gauss never published this result. You must be the first again! Do you have any conjectures as to how many primes have a residue of 0 mod 101? > > a relative preference for 1 and 2 mod 3. By your argument, that means > > that composites should also reflect this relative preference of not > > being 0 mod 3. Yet we know that 1/3 of all numbers are 0 mod 3--so > > clearly, there is another factor at work that is mitigating this preference. > > Uh, yeah, they're divisible by 3. > > > 2. Consider k mod 6. If we exclude multiples of 6--which we know to > > <deleted> > > Yet 6 is NOT prime. You cram more amazing results into one post than most mathematicians into their entire lives, James. > One thing that often fascinates me about arguments about mathematics > is, when it occurs to me that a poster has no interest whatsoever in > what the correct mathematics actually is!!! You actually read your own posts? > And I asked in a thread, why bother doing math research? > > Didn't get much in answer. Not surprised. These people don't care. > > The answer for such people appears to be that they don't bother. But > they like to argue on Usenet for phantom approval, with occasional > words of approval from other people like them who also don't care > about what is correct. Wow. I approve every word you say! Great job, James! > So what happens are endless nonsense arguments because these people > don't give a damn what the mathematics actually says. Keep up your endless arguments, James! > But one day they'll be dead. They will. But YOU will live forever. > Their arguments gone. The emotions that > drive this silly behavior not even a memory. As their bodies turn to > dust, who will care what arguments they made about prime numbers, and > what thoughts drifted through their minds.... I sure envy you with your immortality. > Mathematics is a hard discipline because it ultimately belongs to > those who DO care about what is correct. Who don't give a damn what > people think and don't give a damn what people remember. Will they live forever? > It can only belong to them in that only they can find the truth of > it. For others there's just what people say, or even worse--what they > believe. What do you mean by "they" in plural? Except for you, are there any other such people on Earth? > The searchers just want to know the truth. > > One day maybe someone will invent some juice to inject into the veins > of people who just want to feel important. And they can sigh with > ecstasy in the belief of their math knowledge, and be content--and the > Usenet newsgroups can be quieter. You don't need any juice, James. You derive ecstasy from your own posts to SCM. > But there will be some who will never take such drugs to their veins. > > But they would not live in fantasy now either. What do you mean by "they" in plural? Except for you, are there any other such people on Earth? > Who cares? So some math people are fakes or deluded and are way wrong > but a bunch of other people believed in them. All mathematicans are fakes, except for you, James. > When they're all dead, > who will remember? You will. You will live forever and remember everything.
From: JSH on 22 Jul 2010 10:28
On Jul 21, 9:37 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-22, JSH <jst...(a)gmail.com> wrote: > > > One thing that often fascinates me about arguments about mathematics > > is, when it occurs to me that a poster has no interest whatsoever in > > what the correct mathematics actually is!!! > > Gee, how ironic. I noticed you snipped out the actually relevant > mathematics to embark on your rant. Here it is again: > > When you count the primes less than K with residue 1 vs residue 2 > mod 3, the 2s are in front for almost all values of K. Why is that > if your claim of "no preference" was correct? Why can you have runs of 10 heads in a row when you flip a coin? What does that prove mathematically? Coin bias towards heads? > What *in mathematical language* are you actually claiming? With no preference the primes behave probabilistically with regard to residues modulo a lesser prime. That is the prime residue axiom. Search in ANY search engine: prime residue axiom Runs of any type of behavior are as boring as any supposed pattern in a random sequence. James Harris |