JSH: Random distributions and prime numbers
in [Physics]
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From: JSH on 8 Feb 2010 22:54 On Feb 7, 10:06 am, JSH <jst...(a)gmail.com> wrote: > It's important to highlight the debate about randomness and prime > numbers as for years now the side which claims that random cannot be > found with primes has been winning, when all the evidence actually > says that they can. > > And it's not a minor issue. If primes can give random distributions > then random may possibly defined through prime numbers. Random in our > reality may BE about prime numbers. > > It's an opportunity to answer one of the biggest questions in our > reality: what exactly is random? > > So I presented a rather simple mathematical axiom: > > Prime residue axiom: Given differing primes p_1 and p_2, there is no > preference for any particular residue of p_2 for p_1 mod p_2 over any > other. (And I'll note that I don't consider 0 to be a residue. ) > > The axiom indicates then that by residue, there *should* be random > behavior. Here is an example mod 3. > > Here is what you get with the first 23 primes greater than 3: > > 5 mod 3 = 2, 7 mod 3 = 1, 11 mod 3 = 2, 13 mod 3 = 1, 17 mod 3 = 2, > 19 mod 3 = 1, 23 mod 3 = 2, 29 mod 3 = 2, 31 mod 3 = 1, 37 mod 3 = 1, > 41 mod 3 = 2, 43 mod 3 = 1, 47 mod 3 = 2, 53 mod 3 = 2, 59 mod 3 = 2, > 61 mod 3 = 1, 67 mod 3 = 1, 71 mod 3 = 2, 73 mod 3 = 1, 79 mod 3 = 1, > 83 mod 3 = 2, 89 mod 3 = 2, 97 mod 3 = 1 > > There are some mathematical details which have to be handled though > before you rush to higher primes, as the maximum gap between primes is > roughly p+1, where p^2 is the smallest integer. So to look mod 101, > for instance, you'd need to start at 101^2, before you use primes, so > you'd take the residue modulo primes greater than p^2. > > So I need to clip the first two and start at 11 mod 3. > > So the sequence is > > 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1 > > and by the prime residue axiom, it is random. > > Primes could be used to label random sequences. As imagine the > sequence above were to be labeled, then it could be, residues mod 3, > from 11 through 23. For different random sequences, you could just > look for them in prime residues, and use the primes themselves to > label in the same way. > > The max gap isn't a complicated thing to handle. So if you wish to > test this idea out, you can program it easily enough, and just look at > the distribution with the standard methods to determine randomness. > > But notice, if you did not know about the max gap issue, and did so, > you could convince yourself that the sequence is not random as you'd > have a tendency towards smaller residues until you broke through the > barrier. > > For those still skeptical consider now twin primes. The prime residue > axiom would indicate that for twin primes--two primes in a row > separated only by 2, for instance 11, 13, or 17, 19--the probability > calculation for their occurrence is actually very easy. > > For example, 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 > > (That calculation is fairly straightforward probability.) > > 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. > > However, there is an issue which shifts the probability slightly. > > If you go into the actual residues it jumps out at you: > > 29, 31, 37, 41, 43, 47 > > mod 3: 2, 1, 1, 2, 1, 2 > mod 5: 4, 1, 2, 1, 3, 2 > > Here all the residues for 5 were in evidence so the count came out > right, but for random it should have been possible for ALL the > residues mod 5 to be 4, but it's not because with 6 primes there isn't > enough space in the interval--4*5 = 20, but 49-25=24, where only 12 > are odd and only 6 are primes. So the probability is actually off! A > scenario where all residues are 4 is precluded by the size of the > interval for the larger prime. The prime residue axiom says no residue preference, which leads to a probabilistic argument, but mathematicians have long relied on the slight variation from the simple probability calculation to pretend that there is something non-random going on. But imagine you have a 4 value random number generator, which generates 1 - 4, but you want 6 numbers and make a rule that if you see 4 you will take it only once, and same with 3, but you will allow 1 and 2 as many times as they show. Did you just remove random with that rule? No. > Which is an issue like the max gap problem. > > That will tend to over-count because the higher residues are less > likely to occur because they cannot fit. Easy explanation that jumps > out at you with even a small example. Easy. And what is the impact? About a 12% over count. Hardly enough to justify dismissing random entirely, especially if to do so you have to reverse the prime residue axiom, and assume there IS a preference. For instance 3 has 1 and 2 as its residues. If there is a preference, which one? 1 or 2? It's nonsensical to claim a preference. Lack of a preference is self- evident which is why it's an axiom. > For the smaller primes it's not an issue as if the prime is greater > than interval/(prime count in interval) then that prime isn't affected > and its residues can have purely random behavior. For instance, for 3 > between 25 and 49, you have 24/(6) = 4, and as that is greater than 3, > there is no clipping for 3. So it's like in quantum mechanics if something in the system removes some of the states it doesn't mean that the system no longer has random behavior if it did before. > And that's it. And consider: a simple idea that I call the prime residue axiom that primes have no residue preference leads to an easy proof of the Twin Primes Conjecture--so now it's the Twin Primes Proof--and data already gathered shows the idea to be so effective that it's within 112% of the count at first blush. A deeper look at the over count shows that it's a clipping of certain states at the top end, and you're done. Easy. James Harris
From: harry on 9 Feb 2010 00:56 "JSH" <jstevh(a)gmail.com> wrote in message news:621c070a-4cbf-41f1-970d-e7bb7976a931(a)f17g2000prh.googlegroups.com... On Feb 7, 10:06 am, JSH <jst...(a)gmail.com> wrote: <snip> >And consider: a simple idea that I call the prime residue axiom that >primes have no residue preference leads to an easy proof of the Twin >Primes Conjecture--so now it's the Twin Primes Proof--and data already >gathered shows the idea to be so effective that it's within 112% of >the count at first blush. > >A deeper look at the over count shows that it's a clipping of certain >states at the top end, and you're done. > >Easy. >James Harris so WHERE IS this EASY PROOF ? and 112% within WHAT ? How is the idea EFFECTIVE ? WHat is getting a clipping ?
From: William Hughes on 9 Feb 2010 12:55 On Feb 8, 11:54 pm, JSH <jst...(a)gmail.com> wrote: > The prime residue axiom says no residue preference, which leads to a > probabilistic argument, but mathematicians have long relied on the > slight variation from the simple probability calculation to pretend > that there is something non-random going on. > > But imagine you have a 4 value random number generator, which > generates 1 - 4, but you want 6 numbers and make a rule that if you > see 4 you will take it only once, and same with 3, but you will allow > 1 and 2 as many times as they show. > > Did you just remove random with that rule? Yes, on average you will get more 1's and 2's than 3's and 4's, so your rule leads to a preference for 1's and 2's. - William Hughes
From: apda on 9 Feb 2010 15:37 "JSH" <jstevh(a)gmail.com> wrote in message news:621c070a-4cbf-41f1-970d-e7bb7976a931(a)f17g2000prh.googlegroups.com... On Feb 7, 10:06 am, JSH <jst...(a)gmail.com> wrote: <snip> > So I need to clip the first two and start at 11 mod 3. > > So the sequence is > > 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1 > > and by the prime residue axiom, it is random. WRONG, it is calculated, therefore cannot be random. Psudo-random sort of, but still wrong, where are the 0's ?? <snip> >James Harris
From: Mark Murray on 9 Feb 2010 17:47
On 09/02/2010 20:37, apda wrote: > WRONG, it is calculated, therefore cannot be random. Indeed. > Psudo-random sort of, but still wrong, where are the 0's ?? Get the 0's by subtracting 1. It is a binary sequence. 1/2, 0/1, black/white, its still binary. M -- |