Harristotelian Logic

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.