Harristotelian Logic

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.