Saturday, January 30, 2010
JSH: Twin primes
One of the weirder things I discovered a while back was a resistance to probabilistic explanations for some prime things where the easiest area to see it boldly displayed is with twin primes probability.
To understand fully, imagine that you accept that primes don't have a preferred residue modulo themselves with other primes. For instance, 3 has two potential residues modulo other primes: 1 and 2. Should it prefer 1? Or maybe 2? No. Why would 3 care to lean towards either residue?
If so, then what residue a particular prime has mod 3 should be random.
Ok, so now let's get to twin primes.
Here a trivial little result relating to twin primes 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.
So let's try it out. Between 5² and 7², there are 6 primes. The probability then is given by:
prob = ((5-2)/(5-1))*((3-2)/(3-1) = (3/4)*(1/2) = 0.375
And 6*0.375 =3D 2.25 so you expect 2 twin primes in that interval. The primes are 29, 31, 37, 41, 43, 47 and you'll notice, two twin primes as predicted: 29,31 and 41, 43.
So that's a fun little thing where you can calculate easily when you're bored or something and it works crazy well. Where it is all just about a simple little idea that prime numbers aren't picking in this simple way, and some of you of course know that what I've given looks like a piece of Brun's constant.
Now I noticed that years ago and wondered why math people don't then accept then that it's about probability with twin primes, when they HAVE the probability piece ALREADY in an accepted bit of mathematics, and one answer may be that a simple answer is just not wanted. I found that sad. But it was one of the results that gave me perspective about my other research where I found simple answers and math people wouldn't accept the results as if you look across the research in this area you see a LOT of people with funding to do research in an area where the simple answer means they cannot succeed with anything more complex.
They cannot succeed.
You now know that without having to know complex mathematical ideas! Wow, just like that you're at the top of the field and can shoot down Ph.D's with decades as mathematicians if one of them pretends to produce a twin primes conjecture result.
Given that they cannot succeed they can fund their research indefinitely simply by ignoring the simple answer.
So it's a cash cow.
Oh yeah, so if you figure that twin primes don't care about their residue modulo other primes so they just randomly bounce around by residue then you know the answer to the Twin Primes Conjecture. It's true.
Another way to say it is that prime numbers will never hate p_1 mod p_2 = 2, so that will emerge when p_1 > p_2 simply because the primes don't have a reason to start dropping that possibility, so there will always be twin primes. Easy.
(Um, now though you can also answer Goldbach's Conjecture, and figure out it's false. But unlikely to ever be demonstrated false with an actual counterexample which is sort of a depressing answer I guess.)
So how could academic mathematicians take themselves seriously when they ignore simple answers?
I think it's because of the money. If math is your job and not just a hobby like for me, then simple answers can take away your paycheck. And with that paycheck supporting you and maybe a family with a mortgage, you care more about the paycheck than you do about mathematics.
So it's simple there as well: people paid to do mathematics often cannot be trusted to tell the truth about mathematics if it impacts their paycheck.
I've seen that paid mathematicians routinely lie about mathematics. Routinely lie. As in, it's quite normal for them to make things up completely or avoid simple answers as simple answers don't pay the bills!
And you learn so much just from pondering twin primes and a simple idea.
To understand fully, imagine that you accept that primes don't have a preferred residue modulo themselves with other primes. For instance, 3 has two potential residues modulo other primes: 1 and 2. Should it prefer 1? Or maybe 2? No. Why would 3 care to lean towards either residue?
If so, then what residue a particular prime has mod 3 should be random.
Ok, so now let's get to twin primes.
Here a trivial little result relating to twin primes 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.
So let's try it out. Between 5² and 7², there are 6 primes. The probability then is given by:
prob = ((5-2)/(5-1))*((3-2)/(3-1) = (3/4)*(1/2) = 0.375
And 6*0.375 =3D 2.25 so you expect 2 twin primes in that interval. The primes are 29, 31, 37, 41, 43, 47 and you'll notice, two twin primes as predicted: 29,31 and 41, 43.
So that's a fun little thing where you can calculate easily when you're bored or something and it works crazy well. Where it is all just about a simple little idea that prime numbers aren't picking in this simple way, and some of you of course know that what I've given looks like a piece of Brun's constant.
Now I noticed that years ago and wondered why math people don't then accept then that it's about probability with twin primes, when they HAVE the probability piece ALREADY in an accepted bit of mathematics, and one answer may be that a simple answer is just not wanted. I found that sad. But it was one of the results that gave me perspective about my other research where I found simple answers and math people wouldn't accept the results as if you look across the research in this area you see a LOT of people with funding to do research in an area where the simple answer means they cannot succeed with anything more complex.
They cannot succeed.
You now know that without having to know complex mathematical ideas! Wow, just like that you're at the top of the field and can shoot down Ph.D's with decades as mathematicians if one of them pretends to produce a twin primes conjecture result.
Given that they cannot succeed they can fund their research indefinitely simply by ignoring the simple answer.
So it's a cash cow.
Oh yeah, so if you figure that twin primes don't care about their residue modulo other primes so they just randomly bounce around by residue then you know the answer to the Twin Primes Conjecture. It's true.
Another way to say it is that prime numbers will never hate p_1 mod p_2 = 2, so that will emerge when p_1 > p_2 simply because the primes don't have a reason to start dropping that possibility, so there will always be twin primes. Easy.
(Um, now though you can also answer Goldbach's Conjecture, and figure out it's false. But unlikely to ever be demonstrated false with an actual counterexample which is sort of a depressing answer I guess.)
So how could academic mathematicians take themselves seriously when they ignore simple answers?
I think it's because of the money. If math is your job and not just a hobby like for me, then simple answers can take away your paycheck. And with that paycheck supporting you and maybe a family with a mortgage, you care more about the paycheck than you do about mathematics.
So it's simple there as well: people paid to do mathematics often cannot be trusted to tell the truth about mathematics if it impacts their paycheck.
I've seen that paid mathematicians routinely lie about mathematics. Routinely lie. As in, it's quite normal for them to make things up completely or avoid simple answers as simple answers don't pay the bills!
And you learn so much just from pondering twin primes and a simple idea.
# posted by James Harris @ 14:53 
