Phil Carmody wrote:

> --- jtrjtrjtr2001 <jtrjtrjtr2001@...> wrote:

>> Let p2 be a large prime. We find another large random prime p1, such that

>> y = (2*p1*p2) + 1.

>>

>> Is there any way, one could quantify the probability of y being a prime?

>

> Well, it's just like any arbitrary number of the same size except that

> it's even, it's not divisible by p1, and it's not divisible by p2.

> Therefore there's a prime density boost of (2/1) * (p1/(p1-1)) *

> (p2/(p2-1))

> Those final two factors are effectively 1.

Phil meant y is odd, but another factor must also be considered.

If q is a random odd prime other than p1 and p2, then q does

not divide y-1. q must divide one of the q-1 numbers from

y to y+q-2, and the chance that q divides y increases from

1/q to 1/(q-1).

So the chance that q does not divide changes from

(q-1)/q to (q-2)/(q-1), which corresponds to multiplying the

chance by ((q-2)/(q-1)) / ((q-1)/q) = q*(q-2)/(q-1)^2.

The situation is just like twin primes:

If p is a large prime then p+2 is odd. If q is a random odd

prime other than p then q does not divide p, so the chance

of q dividing p+2 increases from 1/q to 1/(q-1).

The twin prime constant C_2 = 0.66016... is the product

over all odd primes q of q*(q-2)/(q-1)^2

See for example

http://mathworld.wolfram.com/TwinPrimesConstant.html
A random large number n has chance 1/log(n) of being prime.

Our y is odd and also taking C_2 into consideration changes

the chance to 2*C_2/log(y).

--

Jens Kruse Andersen