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

That's quite optmistic. Maybe the one just found was that one!

> > Jack, or someone, any model for when the next one's due,

> > assuming a sperical homogeneous Poisson?

>

> I can't seem to find any "real" numbers on how far the remaining 7

> k values have been tested. The SoB pages seem to indicate that all

> of them have been tested to n=10^7, but that can't be an accurate

> accounting.

>

> However, if we assume that n=10^7 is accurate for each of them,

> there would be about a 50% chance of finding a prime for one of

> the 7 candidate k values with n < 14675000.

>

> That's the good news.

> The bad news is that in order to have a 50% chance of resolving

Remind me to never ignorantly cross you, Jack ;-)

> the Sierpinski conjecture, you'd need to search for all

> n < 2965000000000. (2.965*10^12)

>

> That's an improvement over the last numbers I remember putting

> together, which would have given somewhere around a 40% chance of

> resolving the Sierpinski conjecture by that limit...

>

> http://tech.groups.yahoo.com/group/primenumbers/message/10258

> Still, even a single test in the n > 10^12 range is beyond our

We're most of the way there. Compared with those flipping iron rings at least.

> reasonable capabilities today -- we're just not ready to do

> modular arithmetic on terabit numbers.

Doing it 10^12 times I think will be a harder target.

> Also, the chance to resolve the conjecture before n = 10^9 has

Impressive.

> risen to 3.98% from the previous 2.45%.

> I think a lot of this improvement in the outlook (modest as it

Ah probably the single most pervasive snippet of utter wrongness

> is) is due to the fact that this latest k value (19249) was

> one of the three "toughest" k values to crack of the original

> seventeen (one of the three lowest Proth weights). Only the

> k values of 22699 and 67607 (both still uncracked) have lower

> Proth weights.

I've seen various people throw around on their project fora is

that getting rid of the dense ones is best. Of course, that's

the worse possible situation, you want to get rid of the most

difficult numbers sooner rather than later. I've tried telling

them that, but most just didn't seem to grok the concept.

Thanks for the quick numerics, as always, Jack.

Phil

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http://mail.yahoo.com - Phil Carmody wrote:
>>

First, I realize reading this that I'm being way too precise. With

>> However, if we assume that n=10^7 is accurate for each of them,

>> there would be about a 50% chance of finding a prime for one of

>> the 7 candidate k values with n < 14675000.

>>

>> That's the good news.

>

> That's quite optmistic. Maybe the one just found was that one!

the unknown depth of search, and the inaccuracy of the Proth weight

values, I should probably have just said n < 15_000_000.

In any case, with the remaining 7 k values, if they've been

completely searched up to n < A, we're about 50% to find another

prime with n < 1.5*A. A very quick justification which is very

close to being mathematically "correct":

The remaining 7 k values should produce an aggregate total of about

1.25 primes per "octave" (A < n < 2*A), and the distribution should

be very Poisson-like.

To get a 50% chance of a hit in a Poisson distribution, we need an

expectation of log(2) primes. That requires 0.55 octaves, or

A < n < 1.47*A.

>

If you read the forum link where Louie originally trashed my math,

> Remind me to never ignorantly cross you, Jack ;-)

he admitted very quickly afterward that he did make a mistake and

that I was probably "in the ballpark"...

>

Keeping even a single terabit number in high-speed RAM is far out of

>> Still, even a single test in the n > 10^12 range is beyond our

>> reasonable capabilities today -- we're just not ready to do

>> modular arithmetic on terabit numbers.

>

> We're most of the way there. Compared with those flipping iron rings at least.

> Doing it 10^12 times I think will be a harder target.

the capability of the vast majority of computers in existence -- that's

my point. Clearly we have the capability to build such hardware, but

the whole point of SoB and other cooperative computing projects is to

use inexpensive commonly available PC-like devices, and they're still

many years away from being able to hold even a single terabit number

in RAM.

>

To put things in perspective, the toughest two k values have an

> Ah probably the single most pervasive snippet of utter wrongness

> I've seen various people throw around on their project fora is

> that getting rid of the dense ones is best. Of course, that's

> the worse possible situation, you want to get rid of the most

> difficult numbers sooner rather than later. I've tried telling

> them that, but most just didn't seem to grok the concept.

>

aggregate expectation of 0.15 primes per octave. In very rough

numbers, that means an expectation to find 1 prime between these

two k values as we push n from 10^7 to 10^9 (about 6.6 octaves).

And note of course that 1 prime between those two k values won't

resolve the conjecture. - Now I see the responses to my two questions!

Hurrah for Seveteen or Bust!!

Jean

--- In primenumbers@yahoogroups.com, Jean PennĂ© <jpenne@...> wrote:

>

> Many congrats to the discoverer and all participants for this

> outstanding result!!

>

> However, I have two questions about it :

>

> 1) If I am not wrong, the record has been posted to SoB on March 26 ;

> why is it not yet posted to the top 5000 database ?

>

> 2) What is/are the proving program(s)?

>

> Woul you excuse my curiosity...

>

> Jean

>