17 October 2007

Twin primes

Are There Infinitely Many Primes, by D. A. Goldston, via Casting Out Nines. This was based on a talk given to motivated high school students, so you don't need much background to understand it.

Since this is at least nominally a probability blog, I draw your attention to the following: Call p and p+2 a twin prime pair if they're both prime. Then it's conjectured that the number of twin prime pairs with smaller member less than or equal to x, denoted π2(x), is asymptotically

2 \left[ \prod_{p > 2} \left( 1 - {1 \over (p-1)^2} \right) \right] \int_2^x {1 \over (\log t)^2} \; dt

and there is probabilistic reasoning that leads to this, which is a version of the Twin Prime Conjecture; basically, the "probability" that a number near t is prime is 1/(log t), and so the probability that two numbers near t are both prime should be the square of this. So the number of twin prime pairs under x should be just that integral, except that we have to take some divisiblity information into account, which is what the product out front does. If a number n is odd, then n+2 is twice as likely to be prime as it would ``otherwise" be; if n is not divisible by some odd prime p, then n+2 has ``probability" (p-2)/(p-1) of being not divisible by p (we must have that n isn't two less than a multiple of p), which is 1 - 1/(p-1)2 times the "naive" probability that n+2 isn't divisible by p, namely (p-1)/p.

I rather like results of this form, where we can guess properties of the primes by assuming that the probability that a given integer is divisible by a prime p is 1/p and these events are independent for different primes p; I've talked about this before, when I sketched a heuristic argument for the Goldbach conjecture. Sure,.the primes have no business behaving randomly. But what reason is there that they should behave nonrandomly?

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