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

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

>> 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.

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

And y is not congruent to 1 mod 3, or to 1 mod 5, or to 1 mod 7, etc.

>> Hi,

>>

>> 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.

>

Which should make y somewhat more likely to be divisible by these small

numbers... For instance, half of a random sample of y values should be

divisible by 3.

So I think that the actual density boost would be 2 times the twin prime

constant (~ 0.66) which would make these y numbers about 1.32 times

more likely to be prime than arbitrary numbers of the same size.

Jack - --- Jack Brennen <jb@...> wrote:
> Phil Carmody wrote:

Good catch, Jack!

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

> >> Hi,

> >>

> >> 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.

> >

>

> And y is not congruent to 1 mod 3, or to 1 mod 5, or to 1 mod 7, etc.

Phil

() ASCII ribbon campaign () Hopeless ribbon campaign

/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]

____________________________________________________________________________________

Got a little couch potato?

Check out fun summer activities for kids.

http://search.yahoo.com/search?fr=oni_on_mail&p=summer+activities+for+kids&cs=bz - September 9 jtrjtrjtr2001 wrote:
> y = (2*p1*p2) + 1.

Phil Carmody wrote:

> > Well, it's just like any arbitrary [odd] number of the same size

I wrote:

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

Jack Brennen wrote:

> then q does not divide y-1

> And y is not congruent to 1 mod 3, or to 1 mod 5,

All 4 above posts were mailed September 9 but it took 5 days to

> or to 1 mod 7, etc.

deliver the posts by Jack and I. We are far apart and the posts

showed up the same minute so it seems like a Yahoo problem.

http://tech.groups.yahoo.com/group/primeform/message/8788 was delayed

from August 31 to September 5. Maybe the problem from

http://tech.groups.yahoo.com/group/yg-alerts/message/24 has not been

completely fixed yet. I'm currently posting through

http://tech.groups.yahoo.com/group/primenumbers/ where I haven't seen

a long delay.

--

Jens Kruse Andersen