## Re: [PrimeNumbers] SoB finds 10th prime!

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• ... 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
Message 1 of 8 , May 5, 2007
Phil Carmody wrote:
> --- Paul Underwood <paulunderwood@...> wrote:
>> Congrats to Konstantin Agafonov and Seventeen or Bust for finding the
>> Sierpinski problem prime 19249*2^13018586+1 which has 3.9 million digits!
>
> 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
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...

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.

Also, the chance to resolve the conjecture before n = 10^9 has
risen to 3.98% from the previous 2.45%.

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

Jack
• 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
Message 2 of 8 , May 5, 2007
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
• ... That s quite optmistic. Maybe the one just found was that one! ... Remind me to never ignorantly cross you, Jack ;-) ... We re most of the way there.
Message 3 of 8 , May 6, 2007
--- Jack Brennen <jb@...> wrote:
> Phil Carmody wrote:
> > 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.

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

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

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

> 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.

> Also, the chance to resolve the conjecture before n = 10^9 has
> risen to 3.98% from the previous 2.45%.

Impressive.

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

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.

Thanks for the quick numerics, as always, Jack.

Phil

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• ... First, I realize reading this that I m being way too precise. With the unknown depth of search, and the inaccuracy of the Proth weight values, I should
Message 4 of 8 , May 6, 2007
Phil Carmody wrote:
>>
>> 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!

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

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

If you read the forum link where Louie originally trashed my math,
he admitted very quickly afterward that he did make a mistake and
that I was probably "in the ballpark"...

>
>> 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.

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

>
> 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.
>

To put things in perspective, the toughest two k values have an
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
Message 5 of 8 , May 8, 2007
Now I see the responses to my two questions!

Hurrah for Seveteen or Bust!!

Jean

--- In primenumbers@yahoogroups.com, Jean Penné <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
>
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