Monday, February 15, 2010


JSH: Twin primes probability and blocked states

Remarkably random may have been in plain sight with prime numbers for some time, where a simple probability calculation shows the problem with academic mathematicians interpretations against random which are shown to be self-serving to a "publish or perish" academic mindsight which throws out a quest for the truth.

First off I've noted an area of prime behavior where prime numbers have no reason to have a reason. And that is governed by what I call the prime residue axiom:

Given differing primes p_1 and p_2, where p_1 > p_2, there is no preference for any particular residue of p_2 for p_1 mod p_2 over any other.

There are lots of consequences from this idea but one of the easiest to approach where the mathematical world has a lot of data already calculated is with twin primes.

The prime residue axiom would indicate that for twin primes—two primes in a row separated only by 2, for instance 11, 13, or 17, 19—the probability calculation for their occurrence is actually very easy, as if x is prime and greater than 3 the probability that x+2 is prime is given by:

prob = ((p_j - 2)/(p_j -1))*((p_{j-1} - 2)/(p_{j-1} - 1))*…*(1/2)

where j is the number of primes up to sqrt(x+2), and p_j is the jth prime, p_{j-1} is the prime before it and so forth.

The result is easy as it is just multiplying the probability for each prime that it is NOT true that

x + 2 ≡ 0 mod p

which probability is just the result of dividing one minus the number of non-zero residues by the total number of residues together to get the total probability that a prime plus 2 is also prime.

For example, between 5^2 and 7^2, there are 6 primes. The probability then is given by:

prob = ((5-2)/(5-1))*((3-2)/(3-1) = (3/4)*(1/2) =3D 0.375

And 6*0.375 = 2.25 so you expect 2 twin primes in that interval.

If you look at mathematical literature on this subject you'll notice I have an extraordinary simplification! That is achieved by simply taking the count of primes in the interval while it would seem that mathematicians have wrapped up the prime distribution in their calculation, for instance, since the count of primes is roughly x/ln x, you might achieve something like they have with:

49/ln 49 - 25/ln 25

and the interested reader could see how closely the mainstream math world equations can be approximated with such devices.

So the prime residue axiom leads to a very simple probability calculation which slashes away mountains of complexity from the standard equations on twin primes but it has a problem. It tends to give too high of a probability! So for some reason it indicates more twin primes should occur than tend to occur, which can be seized on as a demonstration of a problem with the idea, though there is good correlation anyway, so it's not, as physics people know, a nail in the coffin of the idea.

But can the over counts be explained? Yes. And trivially by blocked states.

Looking again at the interval from 25 to 49, The primes are 29, 31, 37, 41, 43, 47 and you'll notice, two twin primes as predicted:

29,31 and 41, 43.

But go into the residues by primes—only 3 and 5 matter for the probability calculation—and notice a limit:

29, 31, 37, 41, 43, 47
mod 3: 2, 1, 1, 2, 1, 2
mod 5: 4, 1, 2, 1, 3, 2

For 5, some configurations are NOT possible. And remember the probability is about the residue! The prime residue axiom asserts that the prime number 5 does not care about the residues of the other primes, so you should get random behavior, and in this case you get each of its residues with roughly equal frequency so you get the correct count of twin primes as 3 does the same.

But some states are blocked. For instance, 4, 4, 4, 4, 4, 4 is not possible in that interval for mod 5 because the interval is TOO SMALL for that configuration to fit. So the probability calculation is actually wrong, as it includes that state as a possibility!

Some might think that is significant in ending this idea, except that the assertion simply leads to some simple theory:

If the states are blocked by a size problem, then the effect should diminish with a bigger prime gap.

Notice that 5 ad 7 are themselves twin primes. But 7 and 11 are not, so by that hypothesis, there'd be less impact with the wider gap, which leads to a testable theory:

Over twin primes, if intervals of increasing gap size are considered the over count will tend to decrease, and the overage will tend towards 0.

What I like about this example is how it goes to the question of the quest for truth. I put out ideas about twin primes and probability over 3 years ago. Coming back to the subject I've faced a firestorm of protest on math newsgroups from people vigorously defending the status quo, but one would hope instead that inquisitive minds would be curious about what is the truth.

After all, if random is so trivially demonstrated with twin primes it hardly seems feasible that highly intelligent and dedicated academics would fail to pick up on such a simplification. Doing otherwise would be like physicists ignoring Kepler to keep at Ptolemy's spheres because Kepler's approach was too easy!!!

Here simple ideas lead to simple tests of those ideas, and to forestall claims that I must be the person who performs the verification, I'll note I'm a theoretician, not an experimentalist.

The real question is, who wants answers? Versus, who just wants to argue?

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