## The ultimate target for the gap search [from Riesel]

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• I _love_ Riesel, the best spent \$15 ever! (I d go as far as to say it is worth the Amazon price. After seeing many of the references, Cohen has bubbled up to
Message 1 of 5 , Oct 6, 2001
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I _love_ Riesel, the best spent \$15 ever! (I'd go as far as to say it is worth the Amazon price. After seeing many of the references, Cohen has bubbled up to top the the to-buy list.)

Anyway, here's a pertinent gem, paraphrased a tad,

<<<
[a conjecture by Cramer] would imply that gap Delta_k occurs before
G = O(e^(1.62*sqrt(Delta_k)))
If this holds, a prime free interval of length 1 million ought to be found below e^1620 < 10^704, and enormous number, but nevertheless smaller than many known primes.
>>>

Here are some other targets using the same logic.
Delta_k G
500000 498
200000 315
100000 223
50000 158
20000 100
10000 71
5000 50
2000 32
1000 23

The 1.62 factor comes from the absolute worst detected maximal-gap/position ratio, a ratio that seems to tend towards 1, (though not amazingly quickly).

It can be seen that the lower targets are realistic. Whether hunting for the larger ones is realistic remains to be seen!

Good luck to all those who are hunting, and hats off to Jim for the generous code contribution.

I look forward to seeing Paul's top 20s progress.

Phil

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• Phil wrote that Riesel wrote that Some time back, I
Message 2 of 5 , Oct 6, 2001
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Phil wrote that Riesel wrote that
<<< [a conjecture by Cramer] would imply that gap Delta_k occurs
before G = O(e^(1.62*sqrt(Delta_k))) >>
Some time back, I wrote that Ribenboim wrote that Weintraub wrote
the same thing but with the smaller constant
That's why I said you might get D ~ 137 by 10^70.
David
• ... Date: Wed Sep 26, 2001 3:50 am Subject: Re: 71 digit L=3360 gap http://groups.yahoo.com/group/primenumbers/message/2929 ... Hahah, I was listening.
Message 3 of 5 , Oct 6, 2001
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On Sat, 06 October 2001, d.broadhurst@... wrote:
> Phil wrote that Riesel wrote that
> <<< [a conjecture by Cramer] would imply that gap Delta_k occurs
> before G = O(e^(1.62*sqrt(Delta_k))) >>
> Some time back, I wrote that Ribenboim wrote that Weintraub wrote
> the same thing but with the smaller constant

Date: Wed Sep 26, 2001 3:50 am
Subject: Re: 71 digit L=3360 gap

> That's why I said you might get D ~ 137 by 10^70.
> Glad someone is now listening:-)
> David

Hahah, I was listening. However, you missed out the all important word...

\ /
million
/ \

Really, that was the jaw drop moment! (That coupled with the 'PGFW it quicker than you can say PFGW', 'Martin-ise two before breakfast' 704-digit figure.)

Having said that, I'm most looking forward to the 10^(thirty-something) type records myself. 'small' is beautiful.

Phil

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• Phil Carmody wrote ... ...except that Weintraub would say 500-digit
Message 4 of 5 , Oct 7, 2001
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Phil Carmody wrote
> 704-digit figure
...except that Weintraub would say 500-digit
• ... I d second that to some extent, though much of it is superseded by Crandall and Pomerance. The bits of Cohen that are directly relevant to the
Message 5 of 5 , Oct 8, 2001
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> I _love_ Riesel, the best spent \$15 ever! (I'd go as far as
> to say it is worth the Amazon price. After seeing many of the
> references, Cohen has bubbled up to top the the to-buy list.)

I'd second that to some extent, though much of it is superseded by Crandall
and Pomerance. The bits of Cohen that are directly relevant to the
computational aspect of primes are also well covered in C&P.

Regards,

Paul.

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