Harristotelian Logic

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.