JSH: Twin primes
in [Math]
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From: Mark Murray on 30 Jan 2010 15:51 On 30/01/2010 18:53, JSH wrote: > One of the weirder things I discovered a while back was a resistance > to probabilistic explanations for some prime things where the easiest > area to see it boldly displayed is with twin primes probability. > > To understand fully, imagine that you accept that primes don't have a > preferred residue modulo themselves with other primes. For instance, > 3 has two potential residues modulo other primes: 1 and 2. Should it > prefer 1? Or maybe 2? No. Why would 3 care to lean towards either > residue? (Assertion 1) What happens if I don't accept this? A valid proof would convince me. > If so, then what residue a particular prime has mod 3 should be > random. OK - this result depends on assertion 1 being accepted. Where is the proof of Assertion 1? > Ok, so now let's get to twin primes. .... while depending on Assertion 1. > Here a trivial little result relating to twin primes as if x is prime > and greater than 3 the probability that x+2 is prime is given by: Prove assertion 1 first. M --
From: Joshua Cranmer on 30 Jan 2010 17:06 On 01/30/2010 01:53 PM, JSH wrote: > To understand fully, imagine that you accept that primes don't have a > preferred residue modulo themselves with other primes. For instance, > 3 has two potential residues modulo other primes: 1 and 2. Should it > prefer 1? Or maybe 2? No. Why would 3 care to lean towards either > residue? > > If so, then what residue a particular prime has mod 3 should be > random. Quibbling here, "random" does not mean "uniform." Yes, the residues of primes mod 3 do form a random distribution. But although I don't know what the distribution for residues mod 3 for primes are, I do know you just can't assume it to be the uniform distribution. I'm not a number theorist, but I do seem to recall there being a lot of open questions on the distribution of primes. I would be surprised if the distribution of residues were not one of these. > And 6*0.375 = 2.25 so you expect 2 twin primes in that interval. > The primes are 29, 31, 37, 41, 43, 47 and you'll notice, two twin > primes as predicted: 29,31 and 41, 43. Expected value does not mean actual value. Indeed, if I recall my statistics correctly, deviation from the mean is normally distributed. So, just saying that we "expect two twin primes to be in X interval" doesn't mean "there is two twin primes (or any, for that matter) in X interval." Or I may just be mixing up my statistics stuff. It's been a full year since I've had it, after all, and it's not one of the easy-to-retain bodies of knowledge. > So how could academic mathematicians take themselves seriously when > they ignore simple answers? Probabilistic proofs are generally not interesting, unless they show something to be true with probability 1 or 0. -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
From: MichaelW on 31 Jan 2010 00:28 On Sat, 30 Jan 2010 10:53:17 -0800, JSH wrote: > One of the weirder things I discovered a while back was a resistance to > probabilistic explanations for some prime things where the easiest area > to see it boldly displayed is with twin primes probability. > > To understand fully, imagine that you accept that primes don't have a > preferred residue modulo themselves with other primes. For instance, 3 > has two potential residues modulo other primes: 1 and 2. Should it > prefer 1? Or maybe 2? No. Why would 3 care to lean towards either > residue? > > If so, then what residue a particular prime has mod 3 should be random. > > Ok, so now let's get to twin primes. > > Here a trivial little result relating to twin primes as if x is prime > and greater than 3 the probability that x+2 is prime is given by: > > prob = ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*...*(1/2) > > where j is the number of primes up to sqrt(x+2), and p_j is the jth > prime, p_{j-1} is the prime before it and so forth. > > The result is easy as it is just multiplying the probability for each > prime that it is NOT true that > > x + 2 ≡ 0 mod p > > which probability is just the result of dividing one minus the number of > non-zero residues by the total number of residues together to get the > total probability that a prime plus 2 is also prime. > > So let's try it out. Between 5^2 and 7^2, there are 6 primes. The > probability then is given by: > > prob = ((5-2)/(5-1))*((3-2)/(3-1) = (3/4)*(1/2) = 0.375 > > And 6*0.375 = 2.25 so you expect 2 twin primes in that interval. The > primes are 29, 31, 37, 41, 43, 47 and you'll notice, two twin primes as > predicted: 29,31 and 41, 43. > > So that's a fun little thing where you can calculate easily when you're > bored or something and it works crazy well. Where it is all just about > a simple little idea that prime numbers aren't picking in this simple > way, and some of you of course know that what I've given looks like a > piece of Brun's constant. > > Now I noticed that years ago and wondered why math people don't then > accept then that it's about probability with twin primes, when they HAVE > the probability piece ALREADY in an accepted bit of mathematics, and one > answer may be that a simple answer is just not wanted. I found that sad. > But it was one of the results that gave me perspective about my other > research where I found simple answers and math people wouldn't accept > the results as if you look across the research in this area you see a > LOT of people with funding to do research in an area where the simple > answer means they cannot succeed with anything more complex. > > They cannot succeed. > > You now know that without having to know complex mathematical ideas! > Wow, just like that you're at the top of the field and can shoot down > Ph.D's with decades as mathematicians if one of them pretends to produce > a twin primes conjecture result. > > Given that they cannot succeed they can fund their research indefinitely > simply by ignoring the simple answer. > > So it's a cash cow. > > Oh yeah, so if you figure that twin primes don't care about their > residue modulo other primes so they just randomly bounce around by > residue then you know the answer to the Twin Primes Conjecture. It's > true. > > Another way to say it is that prime numbers will never hate p_1 mod p_2 > = 2, so that will emerge when p_1 > p_2 simply because the primes don't > have a reason to start dropping that possibility, so there will always > be twin primes. Easy. > > (Um, now though you can also answer Goldbach's Conjecture, and figure > out it's false. But unlikely to ever be demonstrated false with an > actual counterexample which is sort of a depressing answer I guess.) > > So how could academic mathematicians take themselves seriously when they > ignore simple answers? > > I think it's because of the money. If math is your job and not just a > hobby like for me, then simple answers can take away your paycheck. And > with that paycheck supporting you and maybe a family with a mortgage, > you care more about the paycheck than you do about mathematics. > > So it's simple there as well: people paid to do mathematics often cannot > be trusted to tell the truth about mathematics if it impacts their > paycheck. > > I've seen that paid mathematicians routinely lie about mathematics. > Routinely lie. As in, it's quite normal for them to make things up > completely or avoid simple answers as simple answers don't pay the > bills! > > And you learn so much just from pondering twin primes and a simple idea. > > > James Harris James, Let's see if I have this correct. Take any two primes in sequence p_1 and p_2 (such that p_1 < p_2 and there is not prime between p_1 and p_2). In your example you have p_1 = 5 and p_2 = 7. If the function pi(n) is the number of primes less than or equal to n then pi(n) will be the same value for all n equal to sqrt(p_1^2) up to sqrt(p_2^2-2). In your example pi(n) is the same for all values of n from sqrt(25) through to sqrt(47) and is 3 (counting {2,3,5} ). (#1) Your function "prob" gives the chance that x+2 is prime given any x is prime as the multiplication of all values of (p_j - 2) / (p_j - 1) where p_j is the j'th prime and j varies from 2 to pi(sqrt(x+2)). Thus for all primes in the range {5^2...7^2} where the pi function always gives a result of 3 (from #1) then the value of prob = .375. (#2) If we know the number of primes between p_1^2 and p_2^2 then we can calculate the expected number of prime pairs by multiplying the number of primes by the prob value from (#2). Since the number of primes between 25 and 49 is 6 then the expected value is 6 * .375 = 2.25; close enough to the actual value of 2. I think this is right and apologies if not. Is there a post on the mymath blog that goes into more detail? Here's my go with larger values assuming I have understood correctly. Between 13^2 and 17^2 there are 22 primes. Since 13 is the 6th prime the prob formula becomes (11/12)*(9/10)*(5/6)*(3/4)*(1/2) = 33/128 (roughly .26). Prob*number of primes ~ 5.7. The actual pairs are (179,181) (191,193) (197,199) (227,229) (239,241) (269,271) (281,283) for a total of 7. This is not as exact a match. How close is the result expected to be? Regards, Michael W.
From: David C. Ullrich on 31 Jan 2010 06:34 On Sat, 30 Jan 2010 10:53:17 -0800 (PST), JSH <jstevh(a)gmail.com> wrote: >One of the weirder things I discovered a while back was a resistance >to probabilistic explanations for some prime things where the easiest >area to see it boldly displayed is with twin primes probability. As discoveries go that's certainly "weird", mainly becasue it's not true. Mathematicians don't "resist" probabilistic arguments about primes; they're big fans of such things. As hints regarding what might be true. Of course they do "resist" the idea that such arguments actually _prove_ things about primes. Because they don't prove anything, because the primes are not actually random in any sense that would make those arguments into actual proofs. >To understand fully, imagine that you accept that primes don't have a >preferred residue modulo themselves with other primes. For instance, >3 has two potential residues modulo other primes: 1 and 2. Should it >prefer 1? Or maybe 2? No. Why would 3 care to lean towards either >residue? > >If so, then what residue a particular prime has mod 3 should be >random. > >Ok, so now let's get to twin primes. > >Here a trivial little result relating to twin primes as if x is prime >and greater than 3 the probability that x+2 is prime is given by: > >prob = ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*...*(1/2) > >where j is the number of primes up to sqrt(x+2), and p_j is the jth >prime, p_{j-1} is the prime before it and so forth. > >The result is easy as it is just multiplying the probability for each >prime that it is NOT true that > >x + 2 ? 0 mod p > >which probability is just the result of dividing one minus the number >of non-zero residues by the total number of residues together to get >the total probability that a prime plus 2 is also prime. > >So let's try it out. Between 5^2 and 7^2, there are 6 primes. The >probability then is given by: > >prob = ((5-2)/(5-1))*((3-2)/(3-1) = (3/4)*(1/2) = 0.375 > >And 6*0.375 = 2.25 so you expect 2 twin primes in that interval. >The primes are 29, 31, 37, 41, 43, 47 and you'll notice, two twin >primes as predicted: 29,31 and 41, 43. > >So that's a fun little thing where you can calculate easily when >you're bored or something and it works crazy well. Where it is all >just about a simple little idea that prime numbers aren't picking in >this simple way, and some of you of course know that what I've given >looks like a piece of Brun's constant. > >Now I noticed that years ago and wondered why math people don't then >accept then that it's about probability with twin primes, when they >HAVE the probability piece ALREADY in an accepted bit of mathematics, >and one answer may be that a simple answer is just not wanted. >I found that sad. But it was one of the results that gave me >perspective about my other research where I found simple answers and >math people wouldn't accept the results as if you look across the >research in this area you see a LOT of people with funding to do >research in an area where the simple answer means they cannot succeed >with anything more complex. > >They cannot succeed. > >You now know that without having to know complex mathematical ideas! >Wow, just like that you're at the top of the field and can shoot down >Ph.D's with decades as mathematicians if one of them pretends to >produce a twin primes conjecture result. > >Given that they cannot succeed they can fund their research >indefinitely simply by ignoring the simple answer. > >So it's a cash cow. > >Oh yeah, so if you figure that twin primes don't care about their >residue modulo other primes so they just randomly bounce around by >residue then you know the answer to the Twin Primes Conjecture. It's >true. > >Another way to say it is that prime numbers will never hate p_1 mod >p_2 = 2, so that will emerge when p_1 > p_2 simply because the primes >don't have a reason to start dropping that possibility, so there will >always be twin primes. Easy. > >(Um, now though you can also answer Goldbach's Conjecture, and figure >out it's false. But unlikely to ever be demonstrated false with an >actual counterexample which is sort of a depressing answer I guess.) > >So how could academic mathematicians take themselves seriously when >they ignore simple answers? > >I think it's because of the money. If math is your job and not just >a >hobby like for me, then simple answers can take away your paycheck. >And with that paycheck supporting you and maybe a family with a >mortgage, you care more about the paycheck than you do about >mathematics. > >So it's simple there as well: people paid to do mathematics often >cannot be trusted to tell the truth about mathematics if it impacts >their paycheck. > >I've seen that paid mathematicians routinely lie about mathematics. >Routinely lie. As in, it's quite normal for them to make things up >completely or avoid simple answers as simple answers don't pay the >bills! > >And you learn so much just from pondering twin primes and a simple >idea. > > >James Harris
From: master1729 on 31 Jan 2010 07:04
David C Ullrich said : > On Sat, 30 Jan 2010 10:53:17 -0800 (PST), JSH > <jstevh(a)gmail.com> > wrote: > > >One of the weirder things I discovered a while back > was a resistance > >to probabilistic explanations for some prime things > where the easiest > >area to see it boldly displayed is with twin primes > probability. > > As discoveries go that's certainly "weird", mainly > becasue it's not > true. > > Mathematicians don't "resist" probabilistic arguments > about primes; > they're big fans of such things. As hints regarding > what might be > true. > > Of course they do "resist" the idea that such > arguments actually > _prove_ things about primes. Because they don't prove > anything, > because the primes are not actually random in any > sense that > would make those arguments into actual proofs. > Indeed. I fully agree. JSH might have a good argument , but no proof. (as i said before ... and btw i do have a proof ) your not a fan of me , but im glad we agree on this. i dont like opponents but at least your not a 'moran' :) regards tommy1729 > >To understand fully, imagine that you accept that > primes don't have a > >preferred residue modulo themselves with other > primes. For instance, > >3 has two potential residues modulo other primes: 1 > and 2. Should it > >prefer 1? Or maybe 2? No. Why would 3 care to > lean towards either > >residue? > > > >If so, then what residue a particular prime has mod > 3 should be > >random. > > > >Ok, so now let's get to twin primes. > > > >Here a trivial little result relating to twin primes > as if x is prime > >and greater than 3 the probability that x+2 is prime > is given by: > > > >prob = ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} > - 1))*...*(1/2) > > > >where j is the number of primes up to sqrt(x+2), and > p_j is the jth > >prime, p_{j-1} is the prime before it and so forth. > > > >The result is easy as it is just multiplying the > probability for each > >prime that it is NOT true that > > > >x + 2 ? 0 mod p > > > >which probability is just the result of dividing one > minus the number > >of non-zero residues by the total number of residues > together to get > >the total probability that a prime plus 2 is also > prime. > > > >So let's try it out. Between 5^2 and 7^2, there are > 6 primes. The > >probability then is given by: > > > >prob = ((5-2)/(5-1))*((3-2)/(3-1) = (3/4)*(1/2) = > 0.375 > > > >And 6*0.375 = 2.25 so you expect 2 twin primes in > that interval. > >The primes are 29, 31, 37, 41, 43, 47 and you'll > notice, two twin > >primes as predicted: 29,31 and 41, 43. > > > >So that's a fun little thing where you can calculate > easily when > >you're bored or something and it works crazy well. > Where it is all > >just about a simple little idea that prime numbers > aren't picking in > >this simple way, and some of you of course know that > what I've given > >looks like a piece of Brun's constant. > > > >Now I noticed that years ago and wondered why math > people don't then > >accept then that it's about probability with twin > primes, when they > >HAVE the probability piece ALREADY in an accepted > bit of mathematics, > >and one answer may be that a simple answer is just > not wanted. > >I found that sad. But it was one of the results > that gave me > >perspective about my other research where I found > simple answers and > >math people wouldn't accept the results as if you > look across the > >research in this area you see a LOT of people with > funding to do > >research in an area where the simple answer means > they cannot succeed > >with anything more complex. > > > >They cannot succeed. > > > >You now know that without having to know complex > mathematical ideas! > >Wow, just like that you're at the top of the field > and can shoot down > >Ph.D's with decades as mathematicians if one of them > pretends to > >produce a twin primes conjecture result. > > > >Given that they cannot succeed they can fund their > research > >indefinitely simply by ignoring the simple answer. > > > >So it's a cash cow. > > > >Oh yeah, so if you figure that twin primes don't > care about their > >residue modulo other primes so they just randomly > bounce around by > >residue then you know the answer to the Twin Primes > Conjecture. It's > >true. > > > >Another way to say it is that prime numbers will > never hate p_1 mod > >p_2 = 2, so that will emerge when p_1 > p_2 simply > because the primes > >don't have a reason to start dropping that > possibility, so there will > >always be twin primes. Easy. > > > >(Um, now though you can also answer Goldbach's > Conjecture, and figure > >out it's false. But unlikely to ever be > demonstrated false with an > >actual counterexample which is sort of a depressing > answer I guess.) > > > >So how could academic mathematicians take themselves > seriously when > >they ignore simple answers? > > > >I think it's because of the money. If math is your > job and not just > >a > >hobby like for me, then simple answers can take away > your paycheck. > >And with that paycheck supporting you and maybe a > family with a > >mortgage, you care more about the paycheck than you > do about > >mathematics. > > > >So it's simple there as well: people paid to do > mathematics often > >cannot be trusted to tell the truth about > mathematics if it impacts > >their paycheck. > > > >I've seen that paid mathematicians routinely lie > about mathematics. > >Routinely lie. As in, it's quite normal for them to > make things up > >completely or avoid simple answers as simple answers > don't pay the > >bills! > > > >And you learn so much just from pondering twin > primes and a simple > >idea. > > > > > >James Harris > |