- 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

Mathematics should not have to involve martyrdom;

Support Eric Weisstein, see http://mathworld.wolfram.com

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http://www.shopping.altavista.com - 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

sqrt(1.165746), instead of 1.62.

That's why I said you might get D ~ 137 by 10^70.

Glad someone is now listening:-)

David - On Sat, 06 October 2001, d.broadhurst@... wrote:
> Phil wrote that Riesel wrote that

Date: Wed Sep 26, 2001 3:50 am

> <<< [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

Subject: Re: 71 digit L=3360 gap

http://groups.yahoo.com/group/primenumbers/message/2929

> sqrt(1.165746), instead of 1.62.

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

> That's why I said you might get D ~ 137 by 10^70.

> Glad someone is now listening:-)

> David

\ /

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

Mathematics should not have to involve martyrdom;

Support Eric Weisstein, see http://mathworld.wolfram.com

Find the best deals on the web at AltaVista Shopping!

http://www.shopping.altavista.com > I _love_ Riesel, the best spent $15 ever! (I'd go as far as

I'd second that to some extent, though much of it is superseded by Crandall

> 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.)

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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