## Randomness debate, some ground rules

So now I'm in a debate about whether or not p mod 3, with p an odd prime greater than 3, is random or not, when I say the result of marching up the primes is random, like coin flips.

That is crucial for my more political claim that mathematicians wrongly ignore randomness that can be found with primes.

But hey, maybe I'm wrong, but what does that mean exactly?

Well, I say primes show no preference for a particular residue modulo a lesser prime, so the behavior is random because no rules are around to make it not be random.

If I am wrong, then that statement is what is wrong, and primes DO show some preference modulo a lesser prime.

To understand what that means consider yet again what I showed with the first 23 primes after 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

So the sequence is

2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1

and if the sequence is random, you can only say that 1 or 2 is next in the sequence.

In case that doesn't make sense, randomness is about NOT knowing what is coming next beyond the 50% probability, like with a coin, can you predict whether it will be heads or tails?

(If you can you can make a lot of money betting people against your ability.)

If there are no rules so that either possibility is equally likely, then either possibility can occur.

That is crucial, if I am wrong, then there are rules that slant whether you get 1 or 2, one way or the other and rules mean PREDICTION is possible.

So someone might be able to say that after the 321st prime, you are actually more likely to get 1 than 2, considering p_321 mod 3.

PREDICTION is key here.

If p mod 3 is NOT random, then knowing rules governing the behavior could allow you to predict, that say, 2 is more likely to be next.

So remember, and this is crucial in this debate, that if I am wrong, then there are some rules that could help you figure out which number—1 or 2—would come next in that sequence.

There can be no other way, logically.

My fear, which is why I'm making yet another post, is that too many of you have been beaten down by the easy tactic that mathematicians and math people have of simply being abstruse.

If things get complicated, people zone out or fear that they're just too stupid to get it.

Well let me be the one who looks stupid here. I'll ask the questions no matter how stupid I look, and notice that people from math circles have no problems with the put-downs.

And neither do I.

The politics here are HUGE. If I can convince some of you that math people routinely lie in this area it could be an immense swing.

If I am wrong, then, oh well, I'll learn something myself about primes. My credibility is not an issue.

Math people have already marked me as a crackpot across the web.

So now you know the stakes. I say math people are vicious and ruthless, quite capable of lying on a huge scale, and of course, they would be mad as hornets and ornery as cornered rattlesnakes for that to come out, so I expect it to get very brutal.

But make no mistake, if p mod 3 is NOT random, then damn it, somebody better start talking about how you predict the next number in the sequence beyond saying there is a 50% chance it is 1, or 2.

Concrete tests are CRUCIAL when dealing with intelligent people who have a lot to lose, and a history of successfully lying to a lot of people all over the world.

## JSH: When nothing works

You put up a label "crackpot" and control the world, as math people demonstrate. Just like Bush and company can control a country people use to think they knew with a word like "liberal" and then do so much more with a word like "terrorist".

People need to belong, need to feel they are right, and hate being the one who doesn't get it.

Math people tell you that you get it—as long as you agree with them accept what they say, including who they label.

I stepped into this a naive young man who assumed wrongly what seems like a long time ago that I could use modern problem solving techniques I learned in school—like brainstorming—against old math problems, and so what if I were wrong a lot?

I believed that whenever I finally got something right, people would accept the truth.

But math people will not.

Some of you may think you know about group-think or how low groups can go, but then you can just ignore information like me managing to get a paper published in a peer reviewed math journal, and math people on the sci.math newsgroup going into a rabid fury upon hearing it, some of them managing an email assault which lead to my paper being unceremoniously yanked, and then later the freaking journal just shut down.

And you know what math people say to all of that?

Not a big deal.

Like it happens all the time.

Easy. And you people eat it up because they have the authority and in the real world it is not about the truth, just like how Bush can control a country with two words: "liberal" and "terrorist".

People need to feel like they're not left out, even if deep down they know they are accepting the lie because everyone else is.

Some of you may know that history belongs to people like me, so that later you will be wondered about, maybe even hated, as other people living in a world that accepts that truth you will not, fight to tell themselves they are not you.

Like none of you would say you would fight Newton, or go along with namecalling against Gauss.

But you would. Yes, you would, if the group went after them, like mathematicians have gone after me.

So why do they do it?

Because they can't do what I can do. They can play with numbers for decades and never make the connection that coin flips are like p mod 3 with primes greater than 3.

And it can seem so unfair, but why worry? Why care, when you can attack the person who can make the discoveries, while you make up stuff in random areas.

They have solved the problem.

People like modern mathematicians and Bush and Republicans in this country have solved the problem of pesky truth—forget it!

Tell people what you want them to believe and use those other modern techniques of psychology, where we know it's not really all that modern.

And some of us know they will fail. But before they fail they will do a lot of damage, destroy a lot of dreams, WASTE A LOT OF PEOPLE'S LIVES AND TIME because they wanted to be something they are not.

Hey, maybe I wish I were an Olympic athlete or could boom out homeruns against a major league pitcher, but I am not an Olympic athlete and probably couldn't manage to connect against a fast-ball.

But what if I went to a ballpark, with my cronies, and people just SAID it was a major league pitcher who threw very soft easy pitches that I barely hit even then, and everyone would cheer really loudly and claim it was a homerun!!!

Don't believe that possible? Why would they?

Because one of them gets to bat next, and the world pays them to play at doing nothing, while pretending to be, someone like me.

You people allow this situation to continue. You bear the responsibility for allowing it to go on.

So you will get the judgement of history.

This time, I paint the mathematicians as the victims.

Your society is what created them, fed them into parasitic behavior, and rewarded them for being vicious people who go after discoverers like me, and otherwise, do nothing of value at all.

## Keeping it simple, primes and randomness

My position is that there is clearly random behavior that can easily be found with prime numbers, but mathematicians can have an endless source for papers and supposed research by ignoring what is mathematically true.

To keep things simple and remove the possibility of any math people stepping in to add unneeded complexity—throwing up sand—let's focus on the simplest area that I've pointed out which is p mod 3, where p is a prime greater than 3, and p mod 3 means to subtract 3 as many times as you can from p without getting a negative number.

What is left over is called the residue. For example 7 mod 3 = 1.

Now then, with the basics established, 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

So the sequence is

2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1

and I say it is a random sequence.

So if 1 were heads and 2 were tails, those could be flips of a coin.

Now consider, if that is true, it is a boon to researchers looking for random lists!

You don't need to flip a coin, just start going up the primes mod 3, and you have perfect randomness.

It could even offer a clue as to how we get randomness in our modern world.

So there is a real benefit beyond the knowledge itself, and a benefit to the sciences as well, where such questions might be answered.

Now it's up to the math people.

My fear is that this idea will mostly just be ignored.

Yup, sure maybe some Usenet poster here or there replies to me to argue, but the bulk of the mathematical community seems to have learned the art of doing nothing.

The challenge to you is to consider that the reason why is that they routinely do nothing of real value, but rely on people not looking at the simple and asking simple questions, while they are awed by people who use a lot of complicated language to do nothing at all of value.

Those in the physics community should know that MOST of the mathematics used to run our technological world was discovered over a hundred years ago.

I suggest to you that many mathematicians today are doing nothing of value at all, and that information can be hidden because of the greatness of what came before—stunning mathematical achievements that are still of great use today, so that few people notice that nothing much new is being done.

This simple area with prime numbers is meant to get people asking questions, and demanding answers versus allowing the modern mathematical community to remain silent when it suits it, only to come out here and there with pronouncements of supposedly great achievements, which then perplexingly show no practical value.

Remember, mathematics is known to be great today because for thousands of years, it has worked in the real world. You could build great things with mathematical ideas that you couldn't or found very difficult to build without them.

## JSH: Math fakes and blind belief

So I am challenging some of you to really consider why you think that modern mathematicians are all doing valuable research, or even mostly doing valuable research.

If you do a search on me, you will find a lot of effort has gone into painting me as a crackpot.

But if I'm right, then of course, mathematicians who are fakes would need to use some namecalling to help hold their position.

But of course, you might think, I'd say that to try and convince, when I'm challenging a very prestigious group of people.

Now way out? Or maybe you should just wait and see which way things fall?

Sure, but there's one problem. The fakes sold the world on a security system that I think can be broken. But breaking it breaks it in a way that people who believe in the fakes can't comprehend, which shatters the illusion of security of a lot of systems.

Your cellphone could be cloned and hacked. Your wireless networks would not be reliable, and you couldn't go to cafes and use your laptop. You couldn't bank on-line, and you couldn't use your credit cards on-line.

I am talking about an economic meltdown so huge you cannot begin to grasp it, because you believe in people who are not what they claim to be.

But can I do it?

Well, if you were me, would you do it?

If instead a few intelligent people were properly skeptical and challenged in just a few ways versus just going along with people who use namecalling as an important tool, then the worst case would not be an issue.

My choice is not to do it, which may seem to be an answer to you, as if I just gave an empty threat.

But I just haven't figured out for certain that I can do it, but I fear that I have opened the door already, but I'm not sure.

I'm not sure partly because I refuse to check, as I contemplate a situation that is outside of anything that I imagined possible.

But if things go badly, I want you to know that you were a major part of the problem because you refused to simply check. Just this simple thing of checking mathematicians on a few key things, like primes and randomness, or why they don't have computers checking math proofs, really.

Revolutions happen because people think they know just enough to do nothing, when action is screaming at them to be taken.

Right now I am the most powerful person on the planet, quite capable of bringing nations to their knees—any nation. And I can say that in the open because a small group of people pretended to be like me, pretended to be capable of making major mathematical discoveries when they couldn't.

So they sold the world on a system that a real mathematician can figure out how to easily break.

And your life depends on what I decide. Whether you believe it or not.

And for now, the decision stands as it has been, to leave just what I have out there, hoping against hope that some one of you will feel an ounce of self-preservation before the hammer falls.

## Playing with random

By doing "research" in areas where randomness dominates, like with twin primes, mathematicians can go on and on, indefinitely—while doing nothing of value at all.

As a racket it is one of the most brilliant ever—as long as people are too dumb to figure it out.

And make no mistake, blurring the lines between what is and is not random with primes can make a huge difference in whether or not some graduate student can do his thesis in a particular area or not, or some tenured professor can get another book out.

It is big business in a small circle of people most of you don't know because let's face it, most people are scared of mathematics!

Making it that much easier.

You live in a world where there is so much information that people have to trust experts.

But what if you made a mistake with that trust? How would you know?

And then again, who really cares?

So you think you know something about prime numbers, and you don't.

So what?

Well, what else to do you think you know?

## JSH: What might people do?

Let's say for the sake of argument that I am right, and mathematicians as a group have seized upon acting brilliant versus actually doing real research, and people found out?

What might they do?

Surprisingly the evidence is actually easy, which I guess is because it is actually hard to be brilliant, so playing at it with mathematics requires that you use simple stuff.

But doesn't there have to be a public will for it as well, where on some level people kind of know it's a con and there can't be that many "beautiful minds" running around, so there must be some fakery?

Doesn't our world society recognize that the likelihood of routine acting at being mathematicians is more likely than there really being that much important research being done?

Like, this stuff is so obvious, let's go deepr tha than just this primes thing, and talk about computer checking of mathematicians. I have heard math people routinely claim computers aren't smart enough!

Or they go on and on about how difficult it would have to be, and who can program such a thing? Or how do you check the programs?

In short, they throw up a lot of sand when MOST people today know that computers seem to be rather good with many things, so why should supposedly logical and mathematically precise statements be beyond checking by computer?

Computers are calling people up on the phone and having conversations with them—very limited yes, but it's hard to accept this supposed inability to check some mathematician on what he claims is a brilliant proof by a computer when they're starting to understand human speech, which is a mess.

Is mathematics logical and orderly, or not?

So get what I'm saying—it's obvious there is something wrong in the field.

Skeptical people understand that fraud happens.

But checks in the mathematical field for published cases of fraud reveal very little if anything.

So are mathematicians the most ethical academics in the history of the world?

Are mathematicians the most ethical academics in the history of the world?

I suggest to you, no, but they are the ones who stand the most to lose from serious scrutiny from a public that wants to believe they are something they are not.

How many great mathematicians in history are there?

Any of you have a clue?

A handful.

That's in all of human history. Yet supposedly there has been an explosion TODAY in mathematical research, and there are thousands of mathematicians working around the world.

I suggest to you that mathematics did NOT get easier.

If you care at all about what you think you knew about primes, consider my post on primes and randomness and then do some reading on what mathematicians say in the same area.

Yes, I am telling you that you don't know what you think you know because a group of people have found a benefit in lying to you.

Oh gee, like that is such a surprise in our modern world.

If it is a surprise to you—you call yourself a skeptic?

## Random prime behavior vs politics

Supposedly there is a debate about how random prime numbers are, but recently I noticed that you could do this simple thing, of considering p mod 3, where p is a prime greater than 3.

So what is p mod 3? It's like clock-time, where when you get to 12, you start over with 1, so for instance 7 mod 3 = 1, and 15 mod 3 = 0, so another way to think about it is, subtract 3 until doing so would give you a negative number, and what's left over is what is called the residue modulo 3.

The residue modulo 3 of 7 is 1. 7 mod 3 = 1.

So it's a simple idea, but if you consider the first 23 primes greater than 3 in order, and look at p mod 3, you get a series:

2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1

And you can check that easily enough as 5 mod 3 = 2, which is why it starts with 2, and as I said above 7 mod 3 = 1, so next is 1 and so forth.

That series is random. There is no rhyme nor reason to it, and given the series, you cannot predict what the next number will be, except to say it will be 1 or 2.

If 1 were heads and 2 tails that could be a series of coin flips.

And why wouldn't it be random? Why should it matter to a prime like 5 what its residue modulo 3 is? Or to 7? Or to greater primes?

You can do the same thing with other primes, and here is p mod 5, with p greater than 5:

2, 1, 3, 2, 4, 3, 4, 1, 2, 1, 3, 2, 3, 4, 1, 2, 1, 3, 4, 3, 4, 2

If you think there is a pattern I suggest you extend it out further. It is a random list.

In general p_1 mod p_2 with p_1 greater than p_2, where both are primes will give you a random list, because in every case the larger primes don't care what their residue modulo the smaller prime is, so you get this easy answer.

But there is a debate within the mathematical community about whether or not primes are random.

So what gives?

Well, consider twin primes. By the standard terminology you have twin primes when one prime is 2 away from another, like 5 and 7 are twin primes, as are 11 and 13.

So you can write twin primes as

p_1 - p_2 = 2

so

p_1 mod p_2 = 2

and I've gone over how primes are random in this area.

That indicates that being twin prime is random and is just about probability, and nothing else.

But if you do a web search on "twin primes" you will read about a lot of research in the area.

How can people do a lot of research on something that is random?

Well, they can if it's not known to be random!!!

You see, mathematicians doing research in this area are doing nothing at all, as the twin primes are random, so there is no reason to them beyond probabilities, like with p mod 3, the probability that it is 1 is 50% as is the probability it is 2.

There is no further information available as the primes don't care about their residue modulo a lesser prime.

With that information, you can get the best answer possible about twin primes—which is simple but I won't go into detail to keep this a simple post—and there is then NO FURTHER INFORMATION POSSIBLE.

So the mathematics in this area should be a couple of paragraphs in some beginning number theory textbook.

But do that web search on "twin primes" and see how much has been written as if there is the possibility of learning more—about something random.

Why would people do such a thing?

Do you have a job? What if your job were being brilliant?

What if to keep your job and live up to the expectations of people around you, you needed to keep being brilliant—or look like you were brilliant—year after year?

Heard of publish or perish?

What better way to keep a career going than to do research on a random process that people don't realize is random?

You can work your entire career—looking brilliant, if they don't know the trick—doing nothing at all.

## Twin primes probability

With my story and my research it is easy to get perspective on just what has to be happening by considering any number of areas, but I'll give you one new one as over the weekend I thought about twin primes, and had a basic simple idea:

If x is prime, then if you loop through all the primes up to sqrt(x+2), and find that none of them divide x+2, then x+2 must be prime, or to put it another way, if

x+2 = 0 mod p

for each prime p up to sqrt(x+2) is NOT true, then x+2 is prime.

Turns out the probability that x+2 is prime relative to a given prime p, is easy as it's just

(p-2)/(p-1)

and you msy be wondering where I get that, when it's just 1 - 1/(p-1) as p-1 is the count of non-zero residues modulo p, like for 5, 1, 2, 3 and 4 are the residues modulo 5. So you take the odds that you get a residue that would mean x+2 is divisible by p, and just subtract that from 1 to get the odds that it will NOT be divisible by p.

And you just do that for each prime up to the sqrt(x+2), and multiply them together to get a probability that if x is prime, x+2 is prime, or, the probability given that x is prime that you have twin primes.

See:
http://mymath.blogspot.com/2006/07/twin-primes-probability_30.html

Now that is so easy you'd think it'd have been part of the mathematical literature a long time ago, but it looks like, scarily, mathematicians just got close, but never quite figured it out, as see

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

where you can see (p-1)/(p-2) in some of the formulas they show, but I haven't seen the simple explanation for why it's there, and it gets worse.

So if x is prime the probability that x+2 is prime is given by

prob = ((p_j - 2)/(p_j - 1))*···*(1/2)

where j is the number of primes up to sqrt(x+2), and p_j is the jth prime.

And if you read my post you'll see I even found a way to relate this to Goldbach's conjecture, where I've just kept going with this REALLY simple idea, as why just consider 2? Why not consider an arbitrary even prime gap g?

Next thing you know, Goldbach's is right in front of you.

But why is ANY of this new??!!!

And can it be ignored? I hope not, but consider all the research I have at this point.

Like, why would they ignore any of my research, like my short proof of Fermat's Last Theorem?

## Prime, probability and denial

So why should I keep bugging the sci.physics newsgroup about prime numbers?

Because mathematicians have done some bizarre crap in this area, and even dragged physics people into some stuff about prime numbers when there are these two systems when it comes to understanding the behavior.

Also, any of you with much training know enough probability and statistics to not only understand how the prime distribution isn't random, while other questions like about twin primes are, but you know there are techniques to determine a random system, which mathematicians can use to settle the question.

So how could they cheat?

Easy. Claim that such techniques should apply to the prime distribution itself i.e. the count of primes when it does not, and then act like that trumps areas where clearly you have randomness like with where you see twin primes.

Smearing the line between the two systems can allow them to confuse people indefinitely, unless you know already the answer, and you figure, hey, these people are going to try to pull something on me.

There are two ways of looking at primes that cover all the ways that primes express themselves in the natural numbers, where one is rigid and determined—not at all random—while the other is completely random.

First I'll show the determined way, which is about the prime distribution itself—that is, the count of primes.

Well, the count of primes up to a given x is exactly determined by a simple calculation using the primes up to and including the square root of x.

For instance, to count the primes up to 24, you need only use

24 - floor(24/2) - floor(24/3) + floor(24/6) + 2 - 1

which is, you subtract the evens, from 24 and then the count of those divisible by 3, and then you add in those divisible by 6—as they've been subtracted twice—and then add in 2 for the primes as 2 got subtracted with the evens and 3 got subtracted with those divisible by 3, and then you subtract one for 1, as one is not prime.

That gives you the EXACT count, and the method is perfect, for any natural x.

In contrast, how many twin primes are there up to 24?

To be a twin prime, given an odd prime x, it must be true in that interval that

x+2

is coprime to 3, as, of course, it will be coprime to 2, but notice, you cannot calculate the count in the same way you could calculate the count of primes!!!

So unlike the count of primes up to 24 the count of twin primes is from a random system.

How do we know it has to be random?

Consider that given primes p_1 mod p_2 if there is a preference for a particular residue, then as the composites are products of the primes that preference would show up in all naturals, which can't happen.

For instance, 3 has 0, 1 and 2 as possible residues, where 0 is impossible for other primes, of course, as for instance 7 is coprime to 3, but notice

7 = 1 mod 3

and what if primes tended to have that residue?

Well if the primes tended to have a residue of 1 modulo 3, then their products would as well, so MOST numbers would be 1 modulo 3, but in fact, we have

1, 2, 3 followed by 4, 5, 6, followed by 7, 8, 9 and so on

showing that the naturals perfectly balance between the three residues.

The primes cannot show a preference for a residue modulo another prime, which is the reason why the difference between primes is random, and you have a random system.

I just explained in a few paragraphs how and why questions about the prime distribution differ from areas that have to do with prime residues modulo other primes, like with the twin primes conjecture, or Goldbach's Conjecture.

Now then, how many mathematicians this year will apply for grants for research on twin primes? Or the prime gap? How many papers could be written in this area?

If you know anything about probability, then see if you can still look at books mathematicians put out in this area the same way, when you understand how SIMPLE it is.

## JSH: That's why insults are used

It's harder and harder to see how mathematicians missed such simple things about prime numbers.

Given the quickness with which math people turn to insults when challenged, my take on it is that on some level they know what they are doing.

But is is just so low, so depressing to think that people could do such things, and then they come after anyone who is in the area who could reveal the truth.

So yeah, as much as mathematicians may hate and fear my work on non-polynomial factorization—enough for a math journal to just keel over and die—they may be just as afraid of my prime counting function, and the possibility of people nosing around and figuring out what is going on.

So what is going on?

By not clearly separating the random system from the determined system with primes, mathematicians can do "research" where they apply tools that can't work to answer questions that are already answered—using probability and statistics.

So, the twin primes conjecture is a probability question, as is Goldbach's conjecture.

At this point, the pattern of postings used against me fit a pattern of political behavior, as I've noted, but why do people resort to political behavior?

It's when the truth doesn't fit their personal needs, so they try to find ways around it, is when people turn to politics.

The insults used against me, the webpages, and all the hostile behavior have been about keeping me from doing what I'm doing now—getting the truth out.

By lying about primes and questions about primes, by NOT using the simple explanations readily available, millions upon millions of dollars in research and prizes could be had by people who did, nothing of importance at all.

And people who might otherwise not be able to work as academics could be professors and teach students, when their "research" was valueless to the world.

I think this problem arose along with the idea of "pure math", probably when the possibility of mathematics as a career arose.

## Understanding primes and randomness

There are two ways of looking at primes that cover all the ways that primes express themselves in the natural numbers, where one is rigid and determined—not at all random—while the other is completely random.

First I'll show the determined way, which is about the prime distribution itself—that is, the count of primes.

Well, the count of primes up to a given x is exactly determined by a simple calculation using the primes up to and including the square root of x.

For instance, to count the primes up to 24, you need only use

24 - floor(24/2) - floor(24/3) + floor(24/6) + 2 - 1

which is, you subtract the evens, from 24 and then the count of those divisible by 3, and then you add in those divisible by 6—as they've been subtracted twice—and then add in 2 for the primes as 2 got subtracted with the evens and 3 got subtracted with those divisible by 3, and then you subtract one for 1, as one is not prime.

That gives you the EXACT count, and the method is perfect, for any natural x.

In contrast, how many twin primes are there up to 24?

To be a twin prime, given an odd prime x, it must be true in that interval that

x+2

is coprime to 3, as, of course, it will be coprime to 2, but notice, you cannot calculate the count in the same way you could calculate the count of primes!!!

So unlike the count of primes up to 24 the count of twin primes is from a random system.

How do we know it has to be random?

Consider that given primes p_1 mod p_2 if there is a preference for a particular residue, then as the composites are products of the primes that preference would show up in all naturals, which can't happen.

For instance, 3 has 0, 1 and 2 as possible residues, where 0 is impossible for other primes, of course, as for instance 7 is coprime to 3, but notice

7 = 1 mod 3

and what if primes tended to have that residue?

Well if the primes tended to have a residue of 1 modulo 3, then their products would as well, so MOST numbers would be 1 modulo 3, but in fact, we have

1, 2, 3 followed by 4, 5, 6, followed by 7, 8, 9 and so on

showing that the naturals perfectly balance between the three residues.

The primes cannot show a preference for a residue modulo another prime, which is the reason why the difference between primes is random, and you have a random system.

I just explained in a few paragraphs how and why questions about the prime distribution differ from areas that have to do with prime residues modulo other primes, like with the twin primes conjecture, or Goldbach's Conjecture.

## Primes issue, significant problem

The problem I outlined in a previous post with the way mathematicians talk about the prime probability is rather huge, though it may be hard to see that at first.

To recap, you can figure out things in an exact way about the prime distribution itself because the naturals are well-ordered, so you can do things like use floor() to get exact counts, like the exact count of naturals that have 3 as a factor up to and including a natural number x is given by floor(x/3).

That is an absolute that works for any prime p, so floor(x/p) is the exact count of naturals up to and including a natural number x that have p as a factor.

So you can get an EXACT prime probability by using simple equations for each prime up to and including the square root of x.

In contast, with the question of values of p_1 mod p_2 where p_1 and p_2 are primes, it is easily shown that the residue of p_1 modulo p_2 CANNOT show any particular preference for a non-zero residue of p_2, so it is random.

And notice there is NO COMPARABLE WAY to get an exact count, to what you can do with counting primes out of the naturals.

So you have TWO SYSTEMS: one where you have exact methods and probability does not apply, and the other where you can't do things exactly and can prove random behavior so only probabilistic methods will work.

But then, there needs to be a rigid separation of approaches to the two different types of problems, but the current math field does not show that separation, and does not admit that probabilistic methods are the only ones that will work in certain areas, like questions on twin primes.

Do some digging if you don't believe me. The prime probability can be related to continuous functions that are well-studied like x/ln x, while in contrast look for similar equations with the twin primes probability, or even more dramatically, for the probability of an arbitrary prime gap g.

The reason this is a huge problem is that there are a lot of people doing research in areas like twin primes with techniques that cannot work, because it is a probabilistic area, and mathematicians have it now as an open question as to whether or not probability applies with primes.

Primes are not THAT mysterious as it's easy to explain behavior that appears to show that probability doesn't work well is just misapplication of probability to an ordered system, and also to prove random behavior like with p_1 mod p_2 in the other areas.

Simple it is mathematically, but politically the impact is immense, as if it is acknowledged it would shift any number of people in various fields out of certain lines of research as they're using methods that cannot succeed.

## JSH: Werid result on the social side

One of the things that supposedly protected from some lone person coming up with remarkable and powerful ideas, yet being ignored, was this supposed thirst for accomplishment from other people, like graduate students, or even undergrads.

Supposedly with the chance to step up and put your names in the history books, some one or more of you would go with the proof, and be willing to stand up against your society.

To date, from what I noticed, only one of you did—for a while—only to give in later.

We're talking about huge amounts of prestige, moving yourself rapidly up the ladder, and best of all, being someone who did a remarkable thing, went with the truth, and helped humanity learn a little more knowledge.

But with all the potential positives, you all ultimately held back, afraid of the negatives.

Terrified of stepping outside of your society and being excoriated, verbally abused for supporting ideas proven mathematicially, and so you all stood together, and the question I wonder about is, why?

Not going with the truth, you get a tremendous amount of pain. Your society gets torn apart.

Many of you who thought you were going to be mathematicians, will never be.

And many of you who call yourselves mathematicians, will soon be in a world which strips you of that label.

So, why?

With so many opportunities, so many times I left things dangling out there, with so many ways you could have grabbed things, done your own research and pushed forward into a brand new world, you ALL chose to go down together.

It is a mystery that tests my knowledge of human psychology. Yes I know about the power of authority figures. I also know about how hard it can be to step outside of your group.

But here, stepping outside could have meant prestige, wealth and fame, and NOT stepping outside could mean huge failure.

The best, simplest conclusion is that all of you concluded that the truth would not be able to win against the social pressure.

And that is remarkable. That is very remarkable. Many of you seem to be overly impressed with social power.

The next step then is to probe why. What circuitry inside your brains causes you ALL to make a very wrong choice, to overly admire or be afraid of something that is so easily broken.

Why aren't any of you logical?

## Prime probability remainder error, fixed

I talked for a while about using the lack of a prime preference for a particular residue modulo another prime i.e. given primes p_1 and p_2

p_1 mod p_2

shows no preference for a particular non-zero residue of p_2 to get prime probability, and this approach was considered to be heuristic, but a couple of days ago I stepped through how it is equivalent to using Legendre's Formula without the use of the floor() function, so errors crop up, which are remainder errors.

But with that explanation in hand, it turns out there is an easy way to correct for the remainder errors which I'll describe in this post.

What's remarkable though is that the research I've been doing does not show that mathematicians figured out the reason for the failure of what they seem to have thought was an intuitive idea—not rigorous—that was to be dropped because it failed, versus seeing the puzzling failure as an opportunity.

Now then consider this approach for the count of primes up to 100:

100*(6/7)*(4/5)*(2/3)*(1/2) = 22.857 to five significant figures

which is using the straightforward but flawed approach of just multiplying and dividing without any use of the floor() function, which adds remainder error.

But here's a simple trick for removing that problem, which is to divide off factors in common between x and p_j*···*2, and then divide by what remains, discarding the remainder as long the result is greater than 1—if it's not then you drop probabilities for the largest prime until it is, and then divide dropping off the remainder—and then multiply that result times (p_k - 1)*···*2, where k is the kth prime remaining and k=j if no primes were dropped, to get the expected.

For example with x=100, first you divide off 10, so what's left is

10*(6/7)*(4)*(2/3)

and since 21 is greater than 10, next you'd drop (6/7), so you have

10*4*(2/3) where now you use floor(), so you have

4*2*floor(10/3)= 4*2*3 = 24

which is only off by 1.

And notice that the the probability given by now dividing by 100, is 24% closer to the exact 25%, as there are 25 primes up to 100.

So there is a simple way to solve that problem.

The reason this idea works is that the actual count of naturals up to and including a natural number x that have a prime p as a factor is given by

floor(x/p)

so absolutely the probability can be said to be floor(x/p)/x. The idea I've outlined is more along the lines of that usage, so it's a simple idea that removes the remainder error.

Intriguingly then, this way of calculating x*(p_j - 1)*···*(1/2) cannot be approximated by using Merten's theorem, as the approximately 12% error that arise from the lack of use of the floor() function is removed.

Here is a recap of the technique written more rigorously:

probPrime(x) = ((p_k - 1)*···*2)*floor(x/(p_k*···*2))/x

such that x>(p_k*···*2), where k is the largest prime less than or equal to j that will fit, where j is the count of primes up to and including sqrt(x).

So shifting things about in a simple way changes everything, and explains the prime distribution without much fuss, while using what people thought was a flawed intuitive approach, when the approach wasn't the problem—the problem was error introduced by not using the floor() function.

This solution is a nice one on a scale that my fellow physics people should be able to appreciate, showing the power of simplicity in thinking, and is I think, an expression of Occam's Razor at work.

Short of it is, simple is good.

## JSH: Why Riemann was likely wrong

I posted a while back that the Riemann Hypothesis was probably wrong, as I'm still not ready to say that I've proven it wrong.

Here's why it's probably wrong.

I noted that Mertens's theorem can be related to prime probability as revealed by a simple idea, and it turns out that the prime number theorem actually traces back for its most important results to only two people: Euler and Chebyshev.

And it is the Euler zeta function.

Mathematicians have progressively written Euler out of the picture till today I'm sure there are young people who would proudly proclaim it Riemann's zeta function, when nope, it's Euler's.

Euler figured out the power series and Chebyshev did some really nice work figuring out the limits which are the prime number theorem.

That is the most important work done in this area.

What Riemann did was try to look back into an imaginary part of the Euler zeta function, and it just logically doesn't make sense that that part has any applicability, especially when you understand the prime probability explanation and why Euler's and Chebyshev's work apply in the first place.

Any of you know about the Mandelbrot set?

You should. And you should know that within the Mandelbrot set there are similar versions of the larger set, all the way to infinity.

I suggest to you that to any extent that the imaginary portion found by analytic continuation of Euler's zeta function appears to match the prime counting function may just be the same kind of thing, where you have something similar to the main set, but not exact.

Besides, if you read Riemann's own words—I read a translation from the German—his approach doesn't sound right as he talks about hoping some stuff balances out over infinity.

Mathematics doesn't work that way. Either things do or they don't, and if they do, it can be proven. No guesswork required.

It's probably not hard to prove it definitively given the connection I've shown with my prime counting function as well as this latest research with prime probability as I've sewn up prime behavior.

But the politics in this area are rather heavy, so I suspect there will still be some mud-slinging from both sides for a while yet.

But for the more intelligent of you, who actually care about what is true, my assessment based on the evidence in front of me now is that the Riemann Hypothesis is false.

## JSH: Easy tests are fun

I like easy tests. This prime probabilty idea is simple to the point of triviality.

It can be tested out easily. It can be related to Mertens's theorem.

If it is part of the literature then fine that can be shown.

If it is not, it should be.

If mathematicians ignore it, why would they?

I can guess at all kinds of reasons, but hopefully that won't be necessary.

And people, I seem to just be warming up. There may not be an area of mathematics I can't touch if it becomes necessary.

So, eventually, you people may be ignoring simplifying results in just about every area, and what will I be doing?

I'll be smiling.

I know you. As I realize that I do know you, I relax as it means I am NOT crazy.

Yup. If enough people call you crazy, you have to start to wonder.

Now I know that you had your reasons that had nothing to do with me, but with my results.

So I can relax and smile, and maybe figure out some other things…

## Mertens's theorem and the prime number theorem

I have been talking for a bit about how simply noting that given a natural x and a simple result about primes, you can talk about prime probability. In this post I'll show how powerful an idea that is by relating it to the prime number theorem and by using Mertens's Theorem.

First off the idea is that given primes p_1 and p_2, there is no preference for

p_1 mod p_2

for any particular non-zero residue of p_2, so that when having some natural x, the probability that x does NOT have p_2 as a factor if p_2 is less than or equal to sqrt(x) is roughly 1 - 1/p, which is (p-1)/p.

Then I can express the probability that a natural x is prime by

probPrime(x) = ((p_j - 1)/p)*···(1/2)

where there are j primes less than or equal to sqrt(x) and p_j is the jth prime.

Now I'll compare results from this idea with what follows from the prime number theorem, and I'll use small examples because they're easier, while it's trivial for others to extend out indefinitely to satisfy themselves and others.

I will use x = 10, x=100 and x=1000.

At x = 10, I have probability of primeness as (2/3)*(1/2) = 1/3, and 10/3 is approximately 3.3.

There are 4 primes in the interval, so 4 - 3 = 1.

Here using the Wikipedia prime number theorem article as a reference.

Li(10) = pi(10) = 2.2

At x=100, I have for my result:

(6/7)*(4/5)*(2/3)*(1/2) = 8/35, and 100(8/35) is approximately 22.9.

There are 25 primes in the interval, so 25 - 22.9 = 2.1.

Li(100) - pi(100) = 5.1.

Finally, at x=1000, I have for my result:

(30/31)*(28/29)*(22/23)*(18/19)*(16/17)*(12/13)*(10/11)*(6/7)*(4/5)*(2/3)*(1/2)

which is approximately 0.153 to three significant figures, so times 1000, I have 153.

There are 168 primes so 168 - 153 = 15.

Li(1000) - pi(1000) = 10.

So it's further off, but not by much. That's as much as I'm going to do by hand. I wonder if anyone else might step in and program this thing and step it out?

Just for fun, how does my idea compare against x/(ln x)?

At x=10, 10/(ln 10) is about 4.3, so it is only about 8% off.

While my idea with 3.3 is about 18% off.

At x=100, 100/(ln 100) is about 21.7, so it is about 13% off.

While my idea with 22.9 is about 8% off.

At x=1000, 1000/(ln 1000) is about 145, so it is about 14% off.

While my idea with 153 is about 9% off.

Mertens's theorem comes into this because as other posters pointed out

((p_j - 1)/p)*···(1/2) approx equals (2*e^{-gamma))/(ln x)

when p_j is again the largest prime up to or equal to sqrt(x).

If my idea holds, and logically it should, then it follows that

(2*e^{-gamma))/(ln x)

approximates the correct value for the probability of primeness as x goes out to infinity.

And that is, oddly enough, possibly new.

If that connection has not been made until now, then no doubt about it people, you are witnessing history, and I'll give the benefit of the doubt, and wait for either someone to cite showing the connection has been made before, or for mathematicians to show appreciation for a beautiful result.

For amateur mathematicians this could be exciting showing what was available still in the way of simple, yet powerful ideas at the very heart of important areas, like core behavior of prime numbers.

## JSH: Nasty and pointless

There are quite a few facts that support my mathematical claims.

But I rarely get to just talk mathematical facts.

Instead I deal with people who learned insults as a tool to control.

Many of you come here looking at the aftermath, listen to them, and think I'm some horrible person who refuses to be reasonable, when the reality is that I am a very disillusioned person who used to think that at least in mathematics all you had to do was figure out something mathematically true that was important.

People who are truly brilliant do not need to call someone a fool or an idiot, a kook or a crackpot, or otherwise work to personally demean another person versus confront them on their ideas.

The problem I think though—which is the horrible theory I think is the best explanatio—is that in math society years ago, a trend started to go for complexity over simple answers as people worried about maintaining careers in a realm that is difficult because mathematical proof is difficult to find.

And we at the end of the devolution of the math field have people who do not do real mathematics, who not only avoid simple answers: they attack people who find them.

They defend a status quo that is anti-discovery.

They work to keep us from knowing.

People, I got a paper published in a mathematical journal. Your newsgroup went after the journal, not just with those emails, but in heated criticisms before them, and you overturned how math society supposedly works, by showing how easily a paper that went through the formal peer review process could just get censored—just like that.

And later the journal died.

Posters rationalize the details, but that was a HUGE event, which the mathematical public managed to just quietly let go by, like the littlest thing, like leaves floating down the river.

But my mathematial discoveries are simplifying results, where I use simple ideas, which could crush the machine that supports so many people—who are doing nothing of value at all.

Hide from the truth if you wish. Rationalize and lie about the details. Tell yourself your society is healthy when its members need the meanest tricks, descend to the depths of the nastiness that human beings can tell each other—to defend it.

But the real world is about development and change.

Humanity did not get to this point, to this level of technological progress, by failing when the discoveries had to be made, and the people who got in the way, well all the ones before, they are history now.

You fight the future, you fight the kind of pressure of progress that brought us planes, trains and computers, and books, and the arts, and medicine and so much more, and you are fighting against the human species itself.

And you will lose.

## JSH: On correcting Ullirch

Someone yet again has made a post about the incident where I made a complaint to Oklahoma State University about a posting by David Ullrich.

The gist of that argument I've gathered is that if you're dealing with a minority, and that person insults you, why shouldn't you be able to use whatever is available to insult them back, including racial slurs?

I think that argument resonates because in America there are people who feel that minorities get a lot of advantages—from being minority—and get away with a lot of things, and like to use race to their advantage—play the "race card".

So, the idea that hey, why can't they get some as well, and why can't others use racial slurs to get back at them, resonated on the newsgroup so that people defended Ullrich.

Trouble is, I said he'd acted as my lapdog in an instance, so there is a problem with equating that to some grievous insult on the level of a racial slur, but also, you have to wonder about the logic behind that argument and why it would resonate.

And some people claimed that it was silly to make a big deal since Ullrich just TALKED of a racial slur being an appropriate reply—without delivering—and how dare I hold anyone on Usenet accountable for anything they say and contact his university?

And they were silent when posters began a rain of racial slurs against me, using the word "nigger" that Ullrich had not said.

The problem here people is consistency—you need to follow rules.

What I see though is a need to have an OTHER, as in someone not like you or in a group you consider your own, to whom you feel anything can be done.

And an US, people who you will defend no matter what, as you see them as your own.

That is the problem that besets our world as when you dehumanize another person, they get upset and dehumanize you, and next thing you're shooting at each other, as why not?

If that person is just some thing, not even human to you, who follows other rules other than human rules, why can't you just FORCE them to act as you want?

If only it were simpler to just say, people are people, that other person may annoy the hell out of me, but hey, that person is still a human being.

When you see people make up rules for themselves that are different than the rules for other peoples or groups, I say, you're seeing why racism is dangerous, as one group decides it is superior, and then they do things that another group can't live with—and why should they?

And next thing you know, they are killing each other.

Some people seem to think the answer is to BE SUPERIOR so that you have the best bombs and the most guns, and you can gun down the weaker group until they accept your assessment that you are superior.

But history shows that does not work.

Your group has an advantage for a while, and later it does not. Your children or grandchildren or later generations pay the price as the pendulum swings, and technology moves without care about your preoccupation with your own grandeur.

Oh, until we get to today, when hey, humanity can just knock itself off completely, and no more worries about who is superior!

## JSH: Clarifying, what is meant by "probability" with primes?

I think there is some confusion about what I mean when I talk about the probability of primes and intervals, which unfortunately may lead some of you to think that I am wrong here.

This thread is a clarifying thread on what I mean when I talk about probability and primes.

The crucial linchpin of the probabilistic approach to considering primes and their behavior is noting that given primes p_1 and p_2,

p_1 mod p_2

gives no preferred non-zero residue, so p_1 has no preferred non-zero residue modulo p_2, which is an absolute easily proven by noting that if primes had preferred residues relative to each other, then since composites are products of primes, they would show a preferred residue as well, but primes and composites together are produced in lock-step order, like,

1, 2, 3, 4, 5 gives you all residues modulo 5, followed by 6, 7, 8, 9, 10, and repeat to infinity

No preference for particular residues modulo 5 over others is allowed by that absolute perfection of the naturals.

It is perfect fairness to all residues. Absolutely perfect out to infinity.

So if I have a natural x, I can look at the potential prime factors less than or equal to sqrt(x), as if x does not have any of those as a factor and it is not 1, then it is prime.

The "probability" then that x has 3 as a factor is 1/3 because there are 3 residues modulo 3 and no expectation that x prefers any one of them. Let's say that the other primes less than or equal to sqrt(x), are 5 and 7, then the probability that x has 5 as a factor is 1/5 and the probability that x has 7 as a factor is 1/7.

Make sense?

But is that the same "probability" given approximately by x/(ln x)?

I thought it was, but it seems Merten's Theorem shows it's not, which was a point brought up to me by a couple of posters.

So how can these ideas be reconciled? The "probability" that x has 3 as a factor is quite naturally 1/3, but that "probability" doesn't match up with the "probability" that follows from using something like x/(ln x), when you use all the primes up to sqrt(x).

Let's look at 1000. At 1000 there are 168 primes that come before. It might make sense to say that the probability of primeness is 168/1000, but in actuality, the probability that the next number is prime is given by considering the probability that it has all the primes up to sqrt(1000) as a factors.

Now consider, the probability that x has 3 as a factor is 1/3, so the probability that it does NOT have 3 as a factor is 2/3, which is just 1- 1/3. Same for each of the primes as in general 1 - 1/p is the probability that p is not a factor. You multiply the probability that p is NOT a factor for each prime, up to 1000, there you have the primes up to 31, so

(30/31)*(28/29)*(22/23)*(18/19)*(16/17)*(12/13)*(10/11)*(6/7)*(4/5)*(2/3)*(1/2)

which is approximately 0.153 to three significant figures

while 1000/(ln 1000) gives approximately 0.145 to three significant figures.

Hmmm…still fairly close, but by Merten's Theorem (if I understand it all correctly) they should start diverging, though the argument that I've given is simple enough that each step can be logically traced out to show absolute perfection so it is a proof—and cannot be wrong.

So what was the objection? Some poster was going on about Merten's and how it showed that the probability found by this method doesn't match up with the prime number theorem.

I've seen at least one poster claiming subtlety in this area, but that's why logic is logic.

A proof is a proof is a proof is a proof.

And proofs are absolutes. Subtlety is in politics, sure, but not in mathematics.

So, the objection from then must be that if you go out far enough, the probability that x is prime diverges from the probability given by the prime number theorem, which I just think is interesting.

So how exactly is that possible?

Maybe the interpretation of Merten's Theorm that I've seen is wrong, or maybe the prime number theorem is wrong, or maybe something else.

I was trained as a physicist, so to me, it's just a matter of figuring out what the answer is, while I think too many mathematicians have sacred cows, and could not begin to contemplate the prime number theorem being wrong.

Some may be giggling by now just at the thought, but why not? In physics overturning results are looked for, and here is a simple argument that I am being told contradicts with what mathematicians think they know from the prime number theorem.

Clarification is definitely in order. But what if, <gasp> the prime number theorem is WRONG?

Can't be? Trust other human beings too much? Think you'd have seen it yourself if it were?

But human beings make mistakes—all the freaking time.

So why not? Wouldn't that be fun?

## JSH: Understanding "churn"

Some of you may not have heard of the word "churn" in the context of people who go through the motions for their own profit, but I think it applicable to what has happened in mathematics with primes.

Simple explanations are ignored for complex ones that allow churn—continual activity writing papers and even books with little or no advancement, which allows more papers and more books.

Churn is when professors aren't actually doing real research, but continually work over areas in primes in ever more complex ways—ignoring simple answers—so that they can write papers and promote their careers.

I've decided that it is the best explanation for mathematicians avoiding simple facts that give simple explanations for the behavior of the prime distribution.

## Primes, probability and politics

It is fairly easy to consider probability with the prime distribution by using one single fact, which I can show with two arbitrary primes p_1 and p_2, and

p_1 mod p_2

with the assertion that there is no preference from p_1 for particular residues of p_2.

For instance, the residues available to primes modulo 3 are 1 and 2. My assertion is that there is no preference for either, so you can have 5 = 2 mod 3, but also have 7 = 1 mod 3, and primes will go either way with 50% probability.

That must be the case as the composites are just products of primes, so if primes, say, favored 1 as a residue modulo 3, then their products would favor 1, and most numbers would be 1 modulo 3, but actually, 1/3 are.

1, 2, 3 is followed by 4, 5, 6 followed by 7, 8, 9, repeat out to infinity…

which is trivial but I show it to emphasize how simple these ideas are, and how rigid they are.

So you can say that given x is prime, the probability that x+2 is divisible by 3 is 1/2.

And you can continue with the other primes up to sqrt(x+2) as it's well-known that if x+2 does not have any of the primes up to sqrt(x+2) as factors, then it must be prime.

If x is not prime, then the probability that x+2 is divisible by 3 is 1/3.

Now the probability that x has p as a factor when p is less than or equal to sqrt(x+2) is just 1/p, by the reasoning above.

So the probability that x does NOT have p as a factor is 1 - 1/p = (p-1)/p, and for each prime you include you just need to multiply the probabilities.

So, trivially, you have the probability that x is prime:

probPrime(x) = ((p_j -1)/p_j)*···*(1/2)

where there are j primes up to sqrt(x) and p_j is the jth prime.

Trivial and easy ideas which follow from noting that primes do not have a preference for residues modulo each other, so why can't you just go out on the web, do some searches, and find that prominently displayed as an early and simple result in research about primes?

You CAN find that mentioned, but you have to dig, and at first I thought it was about a dislike of probability by number theorists, but they talk about probability with 1/(ln x) and primes, why not here?

Well, think about it, what if that is THE explanation for most of the behavior of primes?

Consider that if you think of twin primes, and x as a prime, so you consider x+2, you get a slightly different equation, as for instance, I noted above that if x is prime there is a 1/2 chance that it is 1 modulo 3.

Since primes are coprime to each other, that is, can't have factors in common with each other, so 3 can't be a factor of 5 or any other prime, you drop the 0 residue, so you have p-1 possible residues modulo a prime possible.

So the probability that x+2 is divisible by p when p is less than or equal to sqrt(x+2) is 1/(p-1), so you subtract that from 1 to get the probability that x+2 is NOT divisible by p:

1 - 1/(p-1) = (p-2)/(p-1)

and the probability when x is prime that x+2 is prime is given by

probTwinPrime(x) = ((p_j -2)/p_j - 1)*···*(1/2)

and amazingly, you can see (p - 2)/(p - 1) in the mathematical literature about twin primes!!!

See: http://mathworld.wolfram.com/TwinPrimesConstant.html

So I noticed that and at first I thought maybe I'd just been a bit more brilliant than mathematicians who had just missed something—despite it being easy—but now I wonder.

What if they didn't miss it, but simply didn't want to give the probability link?

Now I'm going to diverge a bit and no this is not really a plug for my own research as I'm going to connect the dots and go into the politics part of this post, as I have other research related to primes, and once in desperation and frustration wrote the first prime counting function article for the Wikipedia in an effort to publicize some research I am certain is important. My final version is in the history of the page:

http://en.wikipedia.org/w/index.php?title=Prime_counting_function&old…

There you can see what I call my prime counting function to distinguish it from the others out there and it has some peculiar features, like it uses a partial difference equation to count prime numbers, which has never been seen before. It uses that partial difference equation with a special constraint which is what forces the exact prime count.

Without that constraint the summation of the partial difference equation with the rest of the equation gives a result close, but not exact to, the prime distribution.

There is a partial differential equation that follows from the partial difference equation, and then, a connection to continuous functions.

I have been talking about that for years. The information I gave before on probability and primes is trivial, and (p-2)/(p-1) is actually visible in mathematical literature you can find on the web, with no mention of the probability argument that I quickly gave, from which it would seem to naturally follow.

Now to politics. I have a question for you, if probability is the crucial feature that controls the observed behavior of primes, and the connection between the prime distribution and continuous functions like 1/(ln x) can all be explained quickly, with a few ideas, like how I've given them in this post, how many careers could be supported by research in the area?

What if the answers to the twin prime conjecture and Goldbach's conjecture and any number of questions about primes could be answered by simple ideas and in a few pages of some number theory text?

How many graduate students would need new theses?

How many new books in this area could be written?

How many professors could get grants or write new papers?

I suggest to you that the politics of the simple answers to the questions about primes has to do with jobs—jobs for mathematicians.

Mathematics is a difficult discipline, but there area LOT of people supposedly mathematicians around the world, but what if they are not really capable of delivering?

Lots of them makes it possible to overwhelm the privileged few with real math talent, while they claim to be what they are not, for the prestige and to make a living.

How many truly gifted people able to do valuable mathematics are there actually in the world?

How many pretenders would it take to create a system of faux work, and activity that really has no value to the world?

Think you'd have too much pride to appear to work feverishly on the Riemann Hypothesis, or the Twin Primes Conjecture, or Goldbach's Conjecture, when simple answers were readily available?

Maybe the greatest lie that modern math people accomplished was in making people believe that mathematical gifts were more common than they are, so that when people who cannot produce overwhelmed the system and took over, they could easily crowd out people who can.

Idle musings? Then go back, look over what I have on prime numbers and consider how easy the mathematics is, and then go out and look over the mathematical literature at the dead zone in this area.

Without the simple explanation, mathematicians worldwide can churn on prime numbers—faking like they're actually doing valuable research.

## JSH: Objectivity

I learned a while back to try and be VERY objective about any mathematical result, because, well, because I had some crappy ideas which I really wanted to be proofs of Fermat's Last Theorem or lead to proofs, which did not.

I remember one day after months of going on and on about some crap ideas, which I had convinced myself were brilliant, I finally realized that posters arguing with me—who I had routinely criticized and insulted for refusing to acknowledge what I thought was a brilliant idea—were right.

Can you imagine that feeling?

I felt horrible, just devastated. And I tried to convince myself I would not do that again.

And then I thought to myself, hey, maybe I just like fiddling with math stuff, and just like to do it without needing anything major, so why not just play around, not talk about it, and do it for its own sake?

So I did, and promptly figured out that weird idea of using

x+y+vz = 0 mod x+y+vz

and there I was again arguing out ideas and complaining about posters lying about the math.

But weird thing, they actually were this time!!!

So you get two things happening, I have crap ideas, which posters say are crap, and they are right, and I have good ideas, which posters STILL say are crap, and they are wrong.

I've learned to try and be objective about the entire thing and see it as an intellectual exercise where part of the solution is politics.

So I make political posts. And I am also starting to fully show some of you just what kind of impact you're looking at here, as your credibility will be shredded with the public having a full and cogent explanation for why "mathematicians" would behave this way.

It turns out you are not as strong as me, so you make up stuff, and can't stand it when it's proven wrong, so you hold on to it, while I did not.

Get it? I've been wrong before too, but I was strong enough to accept what was mathematically true, even when it hurt.

Many of you holding on to failed ideas are just not like me, and my explanation is that mathematics is a DIFFICULT SUBJECT and many of you have learned to lie about it anyway.

So, I have the full explanation for your behavior ready to give to a public that will understand, and contrast you, with me. I had my failures and handled them.

You people, on the other hand, ran away from the truth.

## JSH: They are parasites

The best explanation I can make for the people I keep running up against is that they are parasitic.

Look at how they behave, where insults are ROUTINE. And then they lie about the details so they make it seem like I'm the one who always starts with insults, when I'm responding.

It used to be I'd be slow to respond, trying to work objectively for a while, but these people would run wild with that strategy, so I'm hitting back immediately, reminding people of what it is really like when there can be NO OBJECTIVE discussion because these parasitic individuals come in and muck things up.

If you think this is anything like what it can be, think again.

Things can get much worse. People like this build a lot of negative energy which can simply shut-down any kind of other discussion. And they tend to fight, fight, fight, working together in small teams to try and win.

But I know all their strategies.

Many of you not only allow these people to run a-muck, you often at times cheer them on, but the reason things have not been terribly bad before, has been that I've taken more knocks than I've given out, and tried to be patient at times, but no longer.

I can read between the lines of their posts, as well as remind people that hey, people WHO ACTUALLY ARE EXPERTS IN A FIELD DO NOT NEED TO RESORT TO CHILDISH BEHAVIOR TO MAKE A POINT!!!

And with years of experience posting on Usenet I know that these parasites can be beaten, with their true motivations revealed, they can slowly slink away…

Change is inevitable.

## JSH: Why it's hard to win here

So I have mathematical proofs that cover wide areas of number theory, and hey, even gave a wonderful definition of mathematical proof! But somehow I'm still side-lined with the crank label, as I build up steam and get more math discoveries—while your math professors still just ignore them, or appear to do so from what I've gathered.

How is that possible?

Because they are frauds. Being frauds they have everything to lose by not just going on with business as usual.

There is one sure way to end all of this, which would be for my factoring ideas to be developed into a practical method for factoring that would crash the world economy, and cause a great deal of pain and misery for people who don't give a damn about mathematics.

And they know this.

The other way is attrition, as some of you may have felt proud of what you thought you knew about prime numbers. You might have been excited by the belief that Andrew Wiles actually accomplished something. And you may have spent time anxiously wondering if humanity would ever know if the Riemann Hypothesis were true or not.

But now I keep hammering with mathematical proof that people presenting this information to you are lying, deliberately putting up math-ese, as I call it, so that they can have careers, write books and papers, and act like they are doing something, when they are doing nothing of value at all.

Worse, they come after people who actually accomplish things!!!

So you may have been surprised to find out that John Nash was so snubbed by mathematicians in "pure" math areas, when he got a Nobel prize in Economics. Or you may be surprised to learn that a high school girl isn't worth much notice, supposedly, to the mathematical community so Britney Gallivan should just be so thrilled that at least she gets mentioned a lot in cyberspace, like, they even put her in the Wikipedia, and oh yeah, she even got mentioned on TV!!!

Not a lot for you who were deluded into thinking that if you had some brilliant idea you could become justifiably famous in a good way, in a world that appreciated your intellectual brilliance.

Only to learn now that entire academic fields are taken over by people who are best described as con artists, who MUST come after people who are not—who actually make important discoveries in their fields—to preserve their power.

Yup, it's a sad world with lots of problems, but the good news is that the system always corrects, and I am part of that correction, which is why the fight is so heated: the math wars as I call them are about the future of the world.

Some of you will pick the wrong side, scratch that, MOST of you will pick the wrong side, as it will seem so impossible for me to win from this position, given what has come before, but that's why the future is such a surprise—it's like a box of chocolates, you never know what you're going to get.

with a basic result that I am sure is considered trivial by many, which I found fascinating, which is that given an odd natural number n,

n^2 - 2 = 0 mod p

will only be true for primes p that have 2 as a quadratic residue. Well, duh, you might say, but I realized that offered a neat route to finding large prime numbers, quickly, and since you can use any quadratic residue that is not a square, I'll show what I mean with 17.

And start with n=29, while you can use any natural n to start that is coprime to 17:

29^2 - 17 = 824 = 8*103

And I have the prime number, 103.

So now you use 103^2 - 17 = 32*331, giving you 331.

Then 331^2 - 17 = 8*13693, so I get 13693, in three iterations.

(Astute readers may note that I could use an even number instead of odd numbers with 17 to eliminate the factors of 2, but hey, they're easy to divide off anyway.)

Next is 13693^2 - 17 = 8*19*43*28687, so the next prime is 28687.

Next is 28687^2 - 17 = 16*1429*35993.

One more iteration: 35993^2 - 17 = 32*179*226169

So after 5 iterations starting with 29, the largest prime I have is 226169.

So how is it growing roughly?

One benefit of this approach is that it likes small primes paired with a much larger prime, so even as the numbers get bigger, it's not terribly hard to factor them, as most factors will be small primes.

And that's a simple idea with a simple idea, where you can get randomness by starting with a random or pseudo-random n, and pick whatever quadratic residue you like, as long as it's not a square, and is coprime to n, and off you go, randomly generating ever larger primes.

So yes, you can start with a really large n, and are likely to immediately get much bigger primes than I got starting with n=29.

So what's wrong with this idea which might speed up how quickly your browser can connect by SSL? Well, I'm posting it for objective criticism, but I suspect instead I'll get a lot of namecalling and other juvenile behavior from people who will hate the idea—because I give it.

(So yes, this could be worth money if I cared at all about patenting it, like, millions of dollars U.S. because of the volume of usage of RSA systems.)

That's the real mathematical world. Of course, if it IS practical, then someone somewhere in some country should pick it up, and later I'll use this as yet another example of how real mathematicians behave today, when I say it's because math is a DIFFICULT SUBJECT, so there are more "mathematicians" than can be supported by real discoveries, so the bulk of them make things up!!!

So they have to come after people who don't, which is why they do it in such low ways, like calling names like "crank" or "crackpot" because they are cons trying to keep people from noticing how little they actually accomplish compared to what they claim to accomplish in mathematical areas.

[A reply to someone who wrote the if James does this using a fixed algorithm, then his primes will be predictable.]

Why start with small numbers&helllip;I just gave a quick demonstration of a simple idea.

Practically you'd probably start with a number roughly the size of the prime you wanted, generated pseudo-randomly like with any other method, so it'd probably only take a few iterations.

[A reply to someone who asked James how he proves that he gets prime numbers.]

How does anyone prove primality when looking for primes to use in public keys?

That is detail irrelevant to the idea, which applies to any approach for finding large primes for use with RSA.

An important aspect of this particular approach though is that composites generated by it will tend to pair very small primes with very large primes, making them easy to factor.

### Thursday, August 10, 2006

Here's a roadmap to my research, current as of 8/10/06, the date of this post.

Here's an overview of research that I want to focus on at this point in time. Most of it is on my blog mymath.blogspot.com as this group is meant to be a discussion area for number theory related to what is on the blog, so you can talk about cryptology, basic number theory, algebraic number theory, or any of a variety of subjects near, like computer programming related to number theory, etc. and I hope to do only minimal moderation if necessary, and objective criticisms are appreciated!

I have given a definition of mathematical proof:

http://mymath.blogspot.com/2005/07/definition-of-mathematical-proof.html

Figured out the key properties that define rings that are like the ring of integers:

http://mymath.blogspot.com/2005/03/object-ring.html

Found my own prime counting function, which unlike any other known relies on summing a partial difference equation, which is also why it finds primes on its own, unlike any other known:

http://mymath.blogspot.com/2005/06/counting-primes.html

Fighting mathematicians who have done their best to ignore my research I wrote the first prime counting function article for the Wikipedia, where my latest version is now found in the history of the current page:

http://en.wikipedia.org/w/index.php?title=Prime_counting_function&old…

There readers can see my prime counting function in its fully mathematicized "pure" form, and see how it is a summation, so they can make the leap to understanding how it relates to a partial differential equation and an integration.

I had a paper published in a formally peer reviewed mathematical journal—and then the editors withdrew it after sci.math pressure against it:

http://www.emis.de/journals/SWJPAM/vol2-03.html

Link is to a site mirror as the electronic journal DIED a few months later.

That paper covered some pioneering research advancing modular algebra or the algebra of congruences, extending on the work started by Gauss:

http://mymath.blogspot.com/2005/07/tautological-spaces-factoring.html

Which is a line of attack I used to find a short proof of Fermat's Last Theorem:

http://mymath.blogspot.com/2006/03/proof-of-fermats-last-theorem.html

But I've even considered problems in logic and set theory, handling supposed contradictions:

http://mymath.blogspot.com/2005/06/three-valued-logic.html

and

http://mymath.blogspot.com/2005/05/logical-formedness-axioms.html

and

http://mymath.blogspot.com/2005/06/3-logic-more-basics.html

I have factoring research:

http://mymath.blogspot.com/2006/07/factoring-and-residues.html

And other research on primes specific to probability:

http://mymath.blogspot.com/2006/07/twin-primes-probability_30.html

http://mymath.blogspot.com/2006/08/prime-gap-equation.html

Even some of my minor research is significant, as I talked about a simple way to find primes using quadratic residues:

http://mymath.blogspot.com/2006/04/method-for-quickly-finding-primes.html

The only explanation given the breadth of my research, and dramatic events like a math journal imploding after publishing then retracting a paper of mine is that it is so huge that mathematicians who are living in a political society today—where their word is more important than their research—are fighting a war to deny acceptance of any of it.

If any piece of my research is acknowledged as important from my definition of mathematical proof to my ideas about finding primes then they have to fear that the world will realize what they are doing, so the math wars as I call them are political ones.

It is a fight of group power against mathematical truth.

## Probability and primes

So how do I know my prime probability equations must be valid? How can anyone be sure that this line of research is sound?

First off, why do the primes not show a preference for a particular residue modulo some other prime?

Like, as every prime other than 3 has a residue of 1 or 2 modulo 3, as, for instance, 5 has a residue of 2 modulo 3 since 5-3 = 2, why is there no possibility of preference for a residue of 2 modulo 3 over the residue of 1 modulo 3 which 7 has as 7-6 = 1?

Well, you look at the integers in general and notice that they advance by 1.

So, you have 5, 6, 7, giving you each of the possible residues modulo 3, and then you repeat—8, 9, 10—same pattern..

That goes out to infinity.

It gives no room for preference, as either you have primes or the product of primes, and if the primes could decide something—behave with human stubborness—and try to pick a residue, that pattern out to infinity could not be possible, as, for instance, if primes liked 1 modulo 3, then composites would tend to be 1 modulo 3 versus 2 modulo 3.

Unfortunately mathematicians like to play silly games like talk about "pure" ideas versus practical ones, and probability is considered practical in various areas of human endeavor, including gambling, as well as the sciences, and it seems to me they thought it beneath them to consider probabilistic approaches relying on this intrinsic reality of primes.

And it can seem a bit odd, like, how can there be a probability that 100 is prime?

It's not.

But one can still use this idea over an interval to see how many primes one would expect, as I've demonstrated in posts showing the applicability of this idea, or you might consider some far off interval, or some rather large number and consider that probability.

Yes, it is definite whether or not a number is prime. But that definiteness goes out to infinity.

There are an infinity of primes, and each one of them is definitely a prime, and you will never know most of them.

Importantly, understanding this intrinsic reality of how primes relate to each other—with complete indifference—in terms of residues, allows you to construct logical arguments considering certain problems considered open, like the twin primes conjecture, and Goldbach's conjecture.

Understanding that to the primes it is of no consequence how they line up with regard to residues modulo other primes it is trivial to show that the twin primes conjecture must be true, and Goldbach's conjecture must be false—though counter-examples to his conjecture are unlikely to ever be found as the probability is so low.

That is, they exist, but how do you know where to look? Somewhere out there in infinity there are an infinite number of composites for which Goldbach's simple idea fails, but how do you pick one, and even if you did, is checking it within the range of human ability?

Since the results follow from primes not caring about how they relate to each other by residue, there is no other approach available and no other means of proof one way or the other.

All such attempts, logically, are doomed to fail, as the mathematical reason is just this necessity of prime independence.

Counter arguments are welcome. A proof is not refutable. There is no threat from objective responses questioning the logical chain presented.

I say, the logic is absolute. If that's wrong, prove it. Show a break in the logical chain.

## JSH: Primes and no preference

I have created enough threads that I am bothered by it but I need to emphasize some things that are important as this is a crucial juncture.

First off, why do the primes not show a preference for a particular residue modulo some other prime?

Like, as every prime other than 3 has a residue of 1 or 2 modulo 3, as, for instance, 5 has a residue of 2 modulo 3 since 5-3 = 2, why is there no possibility of preference for a residue of 2 modulo 3 over the residue of 1 modulo 3 which 7 has as 7-6 = 1?

Well, you look at the integers in general and notice that they advance by 1.

So, you have 5, 6, 7, giving you each of the possible residues modulo 3, and then you repeat—8, 9, 10—same pattern..

That goes out to infinity.

It gives no room for preference, as either you have primes or the product of primes, and if the primes could decide something—behave with human stubborness—and try to pick a residue, that pattern out to infinity could not be possible, as, for instance, if primes liked 1 modulo 3, then composites would tend to be 1 modulo 3 versus 2 modulo 3.

Unfortunately mathematicians like to play silly games like talk about "pure" ideas versus practical ones, and probability is considered practical in various areas of human endeavor, including gambling, as well as the sciences, and it seems to me they thought it beneath them to consider probabilistic approaches relying on this intrinsic reality of primes.

And it can seem a bit odd, like, how can there be a probability that 100 is prime? It's not.

But one can still use this idea over an interval to see how many primes one would expect, as I've demonstrated in posts showing the applicability of this idea, or you might consider some far off interval, or some rather large number and consider that probability.

Yes, it is definite whether or not a number is prime. But that definiteness goes out to infinity.

There are an infinity of primes, and each one of them is definitely a prime, and you will never know most of them.

Importantly, understanding this intrinsic reality of how primes relate to each other—with complete indifference—in terms of residues, allows you to construct logical arguments considering certain problems considered open, like the twin primes conjecture, and Goldbach's conjecture.

Understanding that to the primes it is of no consequence how they line up with regard to residues modulo other primes it is trivial to show that the twin primes conjecture must be true, and Goldbach's conjecture must be false—though counter-examples to his conjecture are unlikely to ever be found as the probability is so low.

That is, they exist, but how do you know where to look? Somewhere out there in infinity there are an infinite number of composites for which Goldbach's simple idea fails, but how do you pick one, and even if you did, is checking it within the range of human ability?

Since the results follow from primes not caring about how they relate to each other by residue, there is no other approach available and no other means of proof one way or the other.

All such attempts, logically, are doomed to fail, as the mathematical reason is just this necessity of prime independence.

Counter arguments are welcome. A proof is not refutable. There is no threat from objective responses questioning the logical chain presented.

I say, the logic is absolute. If that's wrong, prove it. Show a break in the logical chain.

## JSH: How ironic

So now I have the explanation for how such simple ideas about primes were not explored—mathematicians thought using probability in this way was beneath them!

Some of it is still weird though, like did no one know where (p-2)/(p-1) came from with twin primes?

See: http://mathworld.wolfram.com/TwinPrimesConstant.html

I still find that bizarre. Tell me someone knew!!! Where did they get it from then?

For those who don't know, I recently started mulling over twin primes and noted the obvious thing that if x were prime if

x+2 = 0 mod p

were NOT true for all primes where p was a prime less than sqrt(x+2) then x+2 had to be prime, so you'd have twin primes, and I worked out the simple formula for the probability given x being a prime that x+2 was prime as well, which is where you see (p-2)/(p-1) in MY research.

I kept going and worked out the probability for when x is a natural that x is prime, and still kept going to figure out the probability that a natural x is a twin prime, and kept going to consider arbitrary even prime gaps.

The mystery has been, how could this research be new? The idea is simple enough, so it seems odd that it hadn't been thought of or pursued before.

Well, apparently mathematicians don't like the idea of thinking of primes with probability in this particular way though they will talk about probability in other ways, preferring thing like 1/ln x, so they just had a disdain for the approach!

I am still hoping to get more info on the particulars, especially how close any particular person got in published research, as there was a citation given on sci.math of someone who did use (p-1)/p for the probability of a natural x not having p as a factor.

The disdain of mathematicians for a logical line of reasoning, however, may have allowed a lot of time spent on questions that can only be answered one way, as the reality is that primes behave in this way where they don't have a preference for a particular residue modulo another prime.

So it can be shown that the twin primes conjecture can be proven only one way, and no other line of argument is available to prove or disprove it.

It can be shown then that Goldbach's conjecture cannot be proven to be true.

There is also then a direct link between the primes and ln x in this fascinating way which shows how prime numbers play a role in ALL of mathematics, including with continuous functions.

The logical arguments necessary to make all those connections are not hard.

Like with twin primes, when none of the primes less than sqrt(x+2) are factors of x+2 when x is a prime, then you have a twin prime, trivial. Otherwise you do not. There is no other reason!!! So it is a probabilistic thing.

Goldbach's conjecture works over a lot of numbers because the probability is extraordinarily high that an even composite C can be written as the sum of two primes.

But there is no other reason.

So over infinity it fails, with an infinity of failures, but with a very low likelihood that anyone will ever find a counterexample.

Regardless of the huge arguments that can erupt over these ideas, there is the prime gap formula itself, which should be part of the literature. So no matter what, if it's not already there, if mathematicians are who many of you still apparently think they are, then they MUST accept that formula and add it to the body of accepted human knowledge.

I fear they will fight. So those of you who give a damn about the truth should get ready as these people will be fighting for their professional lives with their backs against the wall.

I fear they will fight, and they will lose, as at this point in time I have more than enough intellectual tools to crush them at will, but things are likely to get very messy as they bring whatever they can against the truth, and the will of the world.

## JSH: Just consider John Nash

So I am saying that the difficulty of real math discovery was a problem some people decided to solve by making up stuff, and that they come after people who make real discoveries to preserve their system, which to some may seem like grousing over a personal problem with how I have been treated but hey, I have some other people for you to consider.

Like think of John Nash. Famous for winning a Nobel prize in Economics, and there was the movie "A Beautiful Mind", but if you think he has won a mathematical prize where it is a prize just for mathematics as an expression of recognition from his mathematical peers then, I would like you to reply with one.

I hesitate a bit with this one because I worry that maybe I missed one, or possibly since I last checked some mathematicians have decided to recognize him with some pure math prize, as there are some out there.

Last time I checked, he had none.

There are math prizes. Last I checked, unless I missed one, John Nash has none.

In constrast, Andrew Wiles has three, last time I checked.

So why would people who fake things a lot not want to recognize real achievement?

Read up on Britney Gallivan, who made this nifty little discovery when she was in high school. I bring her up and posters reply that hey, she is in MathWorld and the Wikipedia, as if that would be enough for any of them, if they had a significant mathematical accomplishment. Or they mention she was brought up on a TV show!!!

Or read about Anatoly Plotknikov who may have proven that P=NP years ago.

The problem for a system that is about math-ese and public perception that people are making real mathematical accomplishments when they are not is that it is problematic for that system if the bulk of people within it ever shift over to actually being producers who care about being correct—versus appearing to be correct.

Real discoverers are a threat to be handled. And they get handled, because the public does not protect them.

[A reply to someone who told James that John F. Nash and Carlton E. Lemke won the 1978 John von Neumann Theory Prize.]

Are you sure that's a math prize, as in a purely mathematical prize given by mathematicians?

I fear you are just going by the name, as it is includes the name of a mathematician, without checking more thoroughly into the details of who gives it and for what, though at least you put "operations research and management sciences", so were you willfully blind?

I don't think the organization that gives it is a mathematical one, nor am I sure that mathematicians give it, so that example, sadly would not apply, if I'm right.

[A reply to someone who wanted to know whether or not James can categorically state that the problem of factoring efficiently is now definitely solved.]

No. Clearly even if my approach can lead to a solution there would be hurdles to jump in going from the beginning idea to getting something that works practically.

And I have not begun to check it as I have moved on. Doing prime probability now.

My take on it, given where I left the research off at, is that it is scary enough that someone in the world should at least care to check and see, but given what I know about the mathematical community, it's unlikely anyone in the mainstream will check.

That makes it more likely that if it gets checked it will be someone outside of the mainstream.

So that means that if it is possible to develop that idea, and make it practical, it's most likely, in my opinion, that it will be done by a surprise source.

But I did my part, even sending the idea to someone in the U.S. military in the area of cryptography and verifying receipt. If my government is interested in the idea, it sure hasn't told me!

If it's not that can be considered to be a strong opinion that the approach is useless and I'll take it as such, and move on, though when I think about it, I can't quite see why it's useless.

But now I can play with primes!!! I seem to keep coming back to primes, one way or another.

And I'm happier and calmer playing with primes, so no, don't bug me about factoring!

Check out my blog or prior posts. Everything important I have to say on the subject, I've said.

Now, I'm moving on…