## JSH: Twin primes probability and blocked states

Remarkably random may have been in plain sight with prime numbers for some time, where a simple probability calculation shows the problem with academic mathematicians interpretations against random which are shown to be self-serving to a "publish or perish" academic mindsight which throws out a quest for the truth.

First off I've noted an area of prime behavior where prime numbers have no reason to have a reason. And that is governed by what I call the prime residue axiom:

Given differing primes p_1 and p_2, where p_1 > p_2, there is no preference for any particular residue of p_2 for p_1 mod p_2 over any other.

There are lots of consequences from this idea but one of the easiest to approach where the mathematical world has a lot of data already calculated is with 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, 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.

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) =3D 0.375

And 6*0.375 = 2.25 so you expect 2 twin primes in that interval.

If you look at mathematical literature on this subject you'll notice I have an extraordinary simplification! That is achieved by simply taking the count of primes in the interval while it would seem that mathematicians have wrapped up the prime distribution in their calculation, for instance, since the count of primes is roughly x/ln x, you might achieve something like they have with:

49/ln 49 - 25/ln 25

and the interested reader could see how closely the mainstream math world equations can be approximated with such devices.

So the prime residue axiom leads to a very simple probability calculation which slashes away mountains of complexity from the standard equations on twin primes but it has a problem. It tends to give too high of a probability! So for some reason it indicates more twin primes should occur than tend to occur, which can be seized on as a demonstration of a problem with the idea, though there is good correlation anyway, so it's not, as physics people know, a nail in the coffin of the idea.

But can the over counts be explained? Yes. And trivially by blocked states.

Looking again at the interval from 25 to 49, The primes are 29, 31, 37, 41, 43, 47 and you'll notice, two twin primes as predicted:

29,31 and 41, 43.

But go into the residues by primes—only 3 and 5 matter for the probability calculation—and notice a limit:

29, 31, 37, 41, 43, 47
mod 3: 2, 1, 1, 2, 1, 2
mod 5: 4, 1, 2, 1, 3, 2

For 5, some configurations are NOT possible. And remember the probability is about the residue! The prime residue axiom asserts that the prime number 5 does not care about the residues of the other primes, so you should get random behavior, and in this case you get each of its residues with roughly equal frequency so you get the correct count of twin primes as 3 does the same.

But some states are blocked. For instance, 4, 4, 4, 4, 4, 4 is not possible in that interval for mod 5 because the interval is TOO SMALL for that configuration to fit. So the probability calculation is actually wrong, as it includes that state as a possibility!

Some might think that is significant in ending this idea, except that the assertion simply leads to some simple theory:

If the states are blocked by a size problem, then the effect should diminish with a bigger prime gap.

Notice that 5 ad 7 are themselves twin primes. But 7 and 11 are not, so by that hypothesis, there'd be less impact with the wider gap, which leads to a testable theory:

Over twin primes, if intervals of increasing gap size are considered the over count will tend to decrease, and the overage will tend towards 0.

What I like about this example is how it goes to the question of the quest for truth. I put out ideas about twin primes and probability over 3 years ago. Coming back to the subject I've faced a firestorm of protest on math newsgroups from people vigorously defending the status quo, but one would hope instead that inquisitive minds would be curious about what is the truth.

After all, if random is so trivially demonstrated with twin primes it hardly seems feasible that highly intelligent and dedicated academics would fail to pick up on such a simplification. Doing otherwise would be like physicists ignoring Kepler to keep at Ptolemy's spheres because Kepler's approach was too easy!!!

Here simple ideas lead to simple tests of those ideas, and to forestall claims that I must be the person who performs the verification, I'll note I'm a theoretician, not an experimentalist.

The real question is, who wants answers? Versus, who just wants to argue?

## JSH: What puzzles me on twin primes issue

To me the math is easy when it comes to this issue of figuring out when twin primes should occur, but it was easy over three years ago as well.

But I don't get how math people behave on this or other issues. It's like, they have no intellectual curiosity, at all.

But how it that possible?

Sure I can see how competition for grants and a need to write papers could be a big deal, but to get an absolute denial? How?

I try to imagine some mathematician who has said for years he or she wants answers on twin primes and their distribution, who might swear up and down that the answer was what they wanted who could just go weirdly blank if you gave them the solution.

And for those of you who wonder how you can be right, and it not matter, that's how: the people in the field have long since quit caring any more. They've lost that spark. That human curiosity.

I kind of think of them as soulless.

People who have long since quit believing in the search for truth. To them those are just words. They have bills to pay. Status to maintain. There is power in a position. Being a professor is not a bad gig you know. Why should it require that you actually give a damn about your field?

## JSH: But what is right?

For years I've preferred the argumentative style for multiple reasons, where one is that I think that there is a battle of ideas. Correct ideas cannot be defeated so there is no reason for social niceness or jumping through hoops to convince others. Right is right. And wrong is wrong.

However, it occurs to me that others see it differently. And rather than continue in the old style I'm considering evolving my style, and moving away from antagonistic bashings. Those are a lot of fun, but maybe not as productive as I'd like.

So I'm asking the question, what is right?

Years ago I noticed that I started having a lot more fun doing my own amateur musing on mathematics when I lost interest in convincing people, and quit caring if what I found was a really big deal. I'd just work on things for the fun of it. But very soon thereafter I found myself with what looked to me to be the very kind of ideas I'd been looking for before, when I was looking for the BIG IDEA.

So I put them out, and what do you know, ended up right back arguing with people. So it has occurred to me, maybe part of it is, I like to argue! Ok, that self-analysi was easy.

Arguing with people on Usenet, I got the same reactions as before. And to this day of course they say, I didn't change at all! And that NONE of my ideas are really all that important, either being re-hashing of previously known research or just being wrong.

So let's start there. I'll say that I will happily consider that possibility. And I'll note that I do like to argue (but will shy for this thread—I hope—from doing so) and I do like to just play with even simple mathematics for the sheer joy of it, even if I'm doing things already known!

So now what?

I'm not interested in picking up old texts. Not going to engross myself in current literature, and probably won't read many postings (if any) by anyone else. I will not take a math course.

So is there any hope?

What is the answer to the conundrum? Is endless arguing the only possibility? Or must someone give in, and if so, in what way?

There are lots of things I have no interest in doing, but one thing I can do in this thread at least, is try to listen.

Without doubt there will be some nasty replies. I suggest you mostly ignore those, which is what I do. Though I do read some for entertainment value as some can be rather funny. I will try to answer sensible questions, as I see them as being sensible. But may completely ignore people.

Usenet is a freedom of speech area. Posters who maintain that I should just shut-up if I don't follow their rules or some other person's rules are automatically idiots. It is also stupid to request that I just leave the newsgroups as maybe hundreds of people have made that request over more than a decade. You can see how well that has worked for them.

I'm suggesting you use your energy wisely. Moving down old paths that I've seen before will just bore me.

By now I've heard just about every angle a poster can manage.

Try to be creative. Or simply wander off. You need say, nothing at all.

What is right? For discussions. In mathematics. On sci.math. Or alt.math.undergrad.

To me knowledge is not something that is a birthright. It does not matter to me if most of you die in ignorance, or not.

But for your lives, if you do any mathematical work, does it matter if it's wrong?

Does it?

An experiment is in progress: how valuable is your time, really.

If I am right, then the answer from reality is that for many of you, your time is worthless.

Reality does not care if you use time and energy in worthless tasks.

So the experiment is a simple one. I simply watch, and wait. Daily I learn who is worth what, all over the world.

## JSH: Prime gap equation, corrected

My prime residue axiom that primes show no preference by residue, can lead beyond twin primes probability to a general equation for prime gaps.

The prime gap equation given a natural number x and even gap g is

probPrimeGap = ((p_j - 2)/(p_j - 1)*...*(1/2))*(1 - ((p_j - 2)/(p_j-1)*…*(1/2))n*Corr

where n equals (g/2)-1, j is the number of primes up to sqrt(x+g), or sqrt(x-g) if g is negative, p_j is the jth prime, and Corr corrects for any odd primes that are factors of g, if there are any, and if there are none Corr=1, otherwise it adjusts to (p-1)/p for that prime versus (p-2)/(p-1).

Let's try it out. Between 5^2 and 7^2 there are 6 primes. For a prime gap of 4, n=1, and the probability is given then by:

probPrimeGap = ((5 - 2)/(5 - 1))*1/2*(1 - ((5 - 2)/(5 - 1))*1/2) = (3/4)(1/2)(1 - (3/4)(1/2)) = 0.375*0.625 = 0.234375

And 6*0.234375 = 1.40625, so 1 case with a prime gap of 4 is expected.

The primes are 29, 31, 37, 41, 43, 47 and you'll notice, one case with a 4 gap as predicted: 37, 41

As has been noted with my twin primes probability equation the estimate should usually be an over count, but I hypothesize that the effect is less with gaps larger than 2. Notice the equation does default to the twin primes probability equation with g=2.

There is no other known prime gap equation.

## JSH: Quadratic residues and factoring idea

Solving quadratic residues is simply related to factoring.

If you assume f_1 = ak mod p, and f_2 = bk mod p

then easily enough:

k = (a+b)^{-1}(f_1 + f_2) mod p

so if you also have T = f_1*f_2 = abk^2 = abq mod p

you can factor T, when

T = abq mod p, to get f_1 and f_2 and possibly solve for k,

if k^2 = q mod p.

Here also ab and a+b are unique.

## JSH: Random distributions and prime numbers

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.

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.

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.

And that's it.

You have all the information needed to see randomness with prime numbers.

For the residue of one prime relative to another, you have to go beyond the max gap. I've hypothesized that's just a matter of going to primes greater than p^2, to get a random sequence of numbers using that prime's residues. For instance, again for p = 101, you'd use primes greater than 101^2.

I've also shown how you can see the count of twin primes following the predictions from random, with a slight over in the expectation value given by difficulty in fitting in higher residues of the larger primes.

Physicists who are curious who are good with their probability and statistics can test out distributions to see if they now look random, and should consider why they believed before that primes did not give us random.

Primes may have been the key all along. The answer to random. By using them fully we may be able to greatly enhance our understanding of random in our own world.

Who knows? Random in our everyday lives may just be about prime behavior.

### Saturday, February 06, 2010

Posters work hard to make me seem like a bad guy, but mathematics is not about personality.

It also doesn't care about politics. Or have an interest in whether or not you waste your life and energy doing nothing that is correct.

REAL WORLD though, there are people in this world who will take the money for doing something that is wrong.

They just don't care if they're wrong—if they are paid.

Some of you are those kind of people. You don't give a damn if your mathematics is wrong if people think it's right as BELIEF is what you want, not correctness.

Belief can pay the bills as people pay a "researcher" for doing the opposite of their stated job. Paying a person to be wrong, and to fight the truth rather than discover it.

But the world can be kind of dumb. The world can be freaking STUPID. Real math students know that anyway.

They love mathematics because it is beyond human stupidity, when it is correct.

The entire world can hate you. People can line up for days to claim you are wrong. The world can do surveys. Run polls. People can get on talk shows and talk about what an idiot you are. Oprah can do a special on your stupidity.

But if you have a mathematical proof you are right.

You can be right and the entire world be wrong.

For the true math students that is part of the appeal of the field, and why they fight for mathematical proof.

I had to DEFINE it to fight the corruption in the current modern math field! Mathematical proof is defended by real mathematicians.

Everyone else who doesn't defend its value claiming to be a "mathematician" is a pretender.

As then you have the chaos of democracy. You may as well just run an opinion poll to figure out what "proof" is going to be accepted by the current math community, as it is corrupted by people who cannot handle the standard of truth.

For some people truth is a bar that is too high for them to reach.

They lie because they aren't good enough to tell the truth. It's that simple.

Mathematics is a hard discipline. Many will claim the title mathematician, but few can do the real job. So some people cheat.

It's that simple.

It's your life. For some of you accepting reality means you cannot be in the field because you cannot work without cheating. You cannot find the proofs, so you lie.

But for some of you, a life of a lie is no life at all. In the mathematical field or out of it, you wish to be, true.

Then what I offer is the simplest thing of all to you. The joy of discovery. The thrill of being right.

Knowledge beyond human opinion polls. Knowledge that will outlive you, me and the entire human species.

Mathematical proof is your friend. Mathematics loves you when you love the truth not because it has emotions, but because it will stand by you when the entire world stands against you.

And it will win. Always.

## JSH: Prime gap equation

The prime gap equation given a natural number x and even gap g is

probPrimeGap(x) = probPrime(x)*((p_j - 2)/(p_j - 1)*…*(1/3))*(1 - probPrime(x)*((p_j - 2)/(p_j-1)*…*(1/3))n*Corr

where n equals (g/2)-1, j is the number of primes up to sqrt(x+g), or sqrt(x-g) if g is negative, p_j is the jth prime, and Corr corrects for any odd primes that are factors of g, if there are any, and if there are none Corr=1, otherwise it adjusts to (p-1)/p for that prime versus (p-2)/p.

First as shown in my post on twin primes if you have x a prime, then you have the probability that x+2 does NOT have a prime p that is less than or equal to sqrt(x+2) as a factor, is (p-2)/(p-1), but if x may not be prime, you need the probability that it is prime.

You just multiply all the odds of something not happening together to get the odds that none of them happen. So for the first piece of my prime gap equation I just multiplied the probability that x is prime times (p-2)/(p-1) for each prime less than or equal to sqrt(x+g), where for any primes that are factors of g you just have (p-1)/p as there is 0 probability that p can add to another prime and give a number divisible by itself.

For the second piece I simply consider the probability if x is prime that x+2 is NOT prime, and then that x+4 is NOT prime, and so forth up until before the gap, like with g=10, that goes up to x+8, so n=4, as it turns out each probability is roughly the same, and that's how I get the equation, as I just subtract the probability that x+g is prime from 1.

Copied from my math blog post "Prime Gap Equation" of August 9, 2006.

## JSH: How are they still blocking?

Lately I've started throwing everything but the kitchen sink at the problem of these important math results that are being blocked, or I assume they're being blocked as presumably things like mathematical proof of a random distribution are important.

Oh, and Usenet is a test area. I have 3 blogs, and a math group, as well as other outlets for various ideas. I come in on Usenet to try things out, and often mainly to brainstorm, like now.

So why would the world be so slow about picking up things easily demonstrated?

The twin primes result is actually over 3 years old, and you can easily see a probabilistic piece in the twin primes constant that is part of current literature. And aren't there a lot of groups around the world who could use a distribution proven to be perfectly random?

It does puzzle me. Like even the conic parameterization. It's just knowledge. So some people denigrate a set of equations because they call something "Pell's Equation" and consider it only with integers, why does that block hold with the entire world?

And if mathematicians ARE deliberately claiming that random does not exist with primes, by wrapping up the prime distribution with issues like twin primes probability when you can separate it as I've shown, then why would people still trust them?

The dodge is so trivial that the simplification is astounding.

So it'd take some really weak people to even need that dodge, except that there is a lot of money to be made with prime numbers.

But if it's so transparent and there are some very smart people in this world, most of whom aren't part of the cash machine anyway, or hopefully wouldn't care for such a thing anyway, why do they remain silent?

Politics should only work so far. With so many people able to use the mathematics that is clearly seen, there should be a lot of pressure for it to be so used. And don't babble about me being wrong. The twin primes probability example itself is so easy people can play with it on their home computers with a few calculations and watch it work. AND see the key equation within the mathematical literature in the twin primes constant itself. Unbelievable validation, right in front of anyone.

Ok, so brainstorming this issue now. The Usenet experiments continue as I ponder more things to throw on the network, while my main sources have what I call the pure stream. No arguing. No rants or ravings. Just the arguments presented in areas where I have analytics data telling me it's being pumped out to 50+ countries every 30 days.

Today alone my math blog had visits from: Britain, US, Germany, Netherlands, Sweden, Brazil, Poland and Malaysia

Looking at what they were interested in, some clearly weren't math sophisticates, but presumably some were. What do they do when they see the math world as they thought they knew it, shredded down to a world of simple greed and lies? Of politics and academics fighting for grant dollars at the expense of the human race?

Maybe it just turns them off from it all, eh?

Maybe they just get sick inside and decide to think about anything else.

## JSH: Random, politics and money

The twin primes probability result helped me over three years ago when I was still wondering if mathematicians were deliberate frauds or not. It was the result that helped me to conclude that mathematicians knowingly fake it, lie about math, and have no concern about the harm to the world, as random is actually a very valuable concept.

Ok, so what do we know? It's long been established that the prime distribution is NOT random. But a simple idea I call the prime residue axiom allows you to see that primes have random behavior anyway, just not with the distribution.

But if you naively wrap up the question of twin primes probability with the count of prime numbers you can pretend that the count of twin primes has something to do with the prime distribution and is not random.

What I showed—over three years ago—was that if you remove the prime distribution from the picture, you DO find random behavior, and I did that by noting that you find probability from the primes less than the sqrt(x+2), where x+2 is the top of your interval.

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

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.

That probability is the probability that if x+2 does NOT equal 2 mod 3 or 5 as those are the only primes for which you care.

Notice for instance with 29 that 27 does have 3 as a factor. and with 37, 35 has 5 as a factor.

It's not rocket science.

Clearly the probability is determined by the primes less than sqrt(x+2), as to whether or not a particular residue is picked, and that's about probability, and if the primes do not care, it's random.

If you wish to lie though, you can wrap the count of primes here with the probability of primeness and it can appear to be about the prime distribution which is NOT random.

So an easy charlatan's trick is available for the wily mathematician who needs funding.

Now what gives my opinion punch here is that, yup, I noted the random behavior over THREE YEARS AGO.

And posters argued with me back then so you can go back in the record and see!

So your field IS corrupt. By ignoring this easy way to split the prime distribution which is not random from the prime behavior which is, governments around the world have funded fakes to do, nothing of value at all.

I call it, white collar welfare.

So of course I have contempt for the modern math field. No true mathematician would stoop to such tricks, especially with a prized area like prime numbers. But con artists would.

If you are a grad student working on twin primes, then as far as I'm concerned you're a throw away. Either you've been used by others who know, or you know and you're angling to get yours before the world knows.

But random is important in our world. If physicists realized that they had an example of random behavior from prime behavior then they could truly test for random in the real world, as a perfectly random distribution, proven to be so mathematically, would be THE model random distribution.

So you rob the world of untold advances with your greed and stupidity.

But mindless people who would do such a thing would not be expected to care about how devastating they are to humanity, as they are parasites.

You are parasites.

## JSH: Hiding random?

It has long been a matter of debate how random is associated with primes with it well established that the prime distribution itself—the count of prime numbers—is NOT random. But primes may still have random behavior which can be easily seen with twin primes.

To understand though you need to know about probability as well as just a little bit about residues.

Given a prime, like 3, it has two residues, as 0 isn't a residue, so for instance 7 mod 3 =3D 1. The "mod" is clock arithmetic.

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) =3D (3/4)*(1/2) = 0.375

And 6*0.375 =3D 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.

For a bigger interval, here's another example:

Using all the primes up to 100, I get:

prob =3D 0.1558 to 4 significant digits.

So that says that the probability that for a prime between 97^2 and 100^2 that adding 2 to it gives a prime is about 15.58% and there are 66 primes in that interval so there should be about 10 twin primes, and a quick count shows that there are:

(9419, 9421), (9431, 9433), (9437, 9439), (9461, 9463), (9629, 9631), (9677, 9679), (9719, 9721), (9767, 9769), (9857, 9859), (9929, 9931)

So the probability from assuming the primes have no preference for a residue gives you a good estimate and for these examples it was really close but it can wander off as it's a random thing.

The important difference here with what mathematicians do with their own twin primes research is that I separate the prime distribution itself off, by calculating a probability and multiplying that times a known count of primes.

But consider what math people do by looking at the literature, and especially one particular equation. Here's a link to MathWorld:

http://mathworld.wolfram.com/TwinPrimes.html

eqn: (3)

Looking at it, it appears that they tossed in the prime distribution, clearly seen squared, of all things, with x/ln x. And you have the probability formula wrapped up in there:

((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*…*(1/2)

where you can see it by simplifying 1 - 1/(p-1)^2, as that's p(p-2)/(p-1)^2, so you get this excess p/(p-1).

So the equation I give above is part of equations in known literature, as part of what is called the twin primes constant.

But it's obfuscated.

Also by not separating out the count of primes you get a lot of extra which is extraneous.

Now it could be a mistake. But now you know.

So we can just toss off the extra math crap, focus on the random piece and have mathematically a random distribution.

Mathematically perfectly random.

But then the mathematicians lose a few things—some supposedly open problems—so it's an interesting question as to whether or not they'll agree with physicists being practical here!

Regardless, now is available a perfectly random distribution. Mathematically perfectly random, using prime numbers.

The main thing was just to isolate the prime distribution away, and focus on residues of primes relative to each other, and random just jumps out at you.

## JSH: Twin primes literature, Ribenboim

It's weird how mathematicians screwed up twin primes. Looking at MathWorld on their twin primes article I can find the probabilistic formula wrapped up in an equation attributed to Ribenboim:

http://mathworld.wolfram.com/TwinPrimes.html

eqn: (3)

Looking at it, it appears that they tossed in the prime distribution, clearly seen squared, of all things, with x/ln x. And you have the probability formula wrapped up in there:

((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*…*(1/2)

where you can see it by simplifying 1 - 1/(p-1)^2, as that's p(p-2)/(p-1)^2, so you get this excess p/(p-1).

What a nutty mess!

How did that dingbat come up with that crap?

## JSH: Defining random?

I'm posting a lot now because for years I've pondered the issue of random. Some may not know that defining "random" is a HUGE big deal, and years ago I came across clear evidence that random may be based on prime behavior.

If true that can mean a lot for our world.

That prime numbers may give random to the world by residue and them not caring about particular residues is a fascinating implication, if true.

There are physics implications as well but I'm leaving this subject for now on the students newsgroup—as they need to be part of this event—and on sci.math itself.

Years to ponder. I'm lucky. I get to think about things for quite a while before I move forward.

Now is the time to move forward with the prime residue axiom and the possible answer to one of the greatest questions human beings have ever raised:

What is random? 