Harristotelian Logic

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?