## GAP of 93918 between two PRPs

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• Hello, I have recently found that there are no primes between the two PRPs 26993#-36017 and 26993#+57901. It s easy to show that generaly if p(n)#+1 and
Message 1 of 4 , Aug 31, 2001
Hello,

I have recently found that there are no primes between the two
PRPs 26993#-36017 and 26993#+57901.
It's easy to show that generaly if p(n)#+1 and p(n)#-1
are composite (not rare), the gap between p(n)#-a and p(n)#+b
which are PRPs is larger than 2*p(n+1), and if a and b < p(n+1)^2,
a and b can only be primes.

Henri Lifchitz
Paris, France

[Non-text portions of this message have been removed]
• Using the log(p):g ratio, this comes in at about 0.287, which is better than Milton s but not as good as Jose Luis . Of course, purely on a size basis it is
Message 2 of 4 , Sep 3, 2001
Using the log(p):g ratio, this comes in at about 0.287, which is
better than Milton's but not as good as Jose Luis'. Of course, purely
on a size basis it is the new record.

Joe.

P.S. I'm going to try to keep a top ten (or however many) list of
these composite runs. Note that these are not primes gaps unless the
endpoints are proved prime, as in the Dubner case.

Subject: [PrimeNumbers] GAP of 93918 between two PRPs
Author: "Henri LIFCHITZ" <HLifchitz@...> at Internet
Date: 31/08/01 18:34

Hello,

I have recently found that there are no primes between the two
PRPs 26993#-36017 and 26993#+57901.
It's easy to show that generaly if p(n)#+1 and p(n)#-1
are composite (not rare), the gap between p(n)#-a and p(n)#+b
which are PRPs is larger than 2*p(n+1), and if a and b < p(n+1)^2,
a and b can only be primes.

Henri Lifchitz
Paris, France

[Non-text portions of this message have been removed]

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• They are minimal prime gaps (the gaps can only get larger if either end point is proved not to be prime). Also they are prime gaps with probabilty 99.999.. %.
Message 3 of 4 , Sep 3, 2001
They are minimal prime gaps (the gaps can only get larger
if either end point is proved not to be prime).

Also they are prime gaps with probabilty 99.999.. %.

Where do you keep this list?

Milton L. Brown
miltbrown@...

----- Original Message -----
From: <joe.mclean@...>
Sent: Monday, September 03, 2001 3:16 AM
Subject: Re: [PrimeNumbers] GAP of 93918 between two PRPs

> Using the log(p):g ratio, this comes in at about 0.287, which is
> better than Milton's but not as good as Jose Luis'. Of course, purely
> on a size basis it is the new record.
>
> Joe.
>
> P.S. I'm going to try to keep a top ten (or however many) list of
> these composite runs. Note that these are not primes gaps unless the
> endpoints are proved prime, as in the Dubner case.
>
>
_________________________________
> Subject: [PrimeNumbers] GAP of 93918 between two PRPs
> Author: "Henri LIFCHITZ" <HLifchitz@...> at Internet
> Date: 31/08/01 18:34
>
>
> Hello,
>
> I have recently found that there are no primes between the two
> PRPs 26993#-36017 and 26993#+57901.
> It's easy to show that generaly if p(n)#+1 and p(n)#-1
> are composite (not rare), the gap between p(n)#-a and p(n)#+b
> which are PRPs is larger than 2*p(n+1), and if a and b < p(n+1)^2,
> a and b can only be primes.
>
> Henri Lifchitz
> Paris, France
>
>
> [Non-text portions of this message have been removed]
>
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
> The Prime Pages : http://www.primepages.org
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>
>
> --------------------------------------------------------------------------
--
> Disclaimer:
> This message is intended only for use of the addressee. If this message
> was sent to you in error, please notify the sender and delete this
message.
> Glasgow City Council cannot accept responsibility for viruses, so please
> scan attachments. Views expressed in this message do not necessarily
reflect
> those of the Council who will not necessarily be bound by its contents.
>
> --------------------------------------------------------------------------
--
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
> The Prime Pages : http://www.primepages.org
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
• ... As Paul J. pointed out, size can be taken arbitrarily far if one looks at probable primes. I don t think it s a target that is worthwhile pushing people
Message 4 of 4 , Sep 3, 2001
On Mon, 03 September 2001, joe.mclean@... wrote:
> P.S. I'm going to try to keep a top ten (or however many) list of
> these composite runs. Note that these are not primes gaps unless the
> endpoints are proved prime, as in the Dubner case.

As Paul J. pointed out, size can be taken arbitrarily far if one looks at probable primes. I don't think it's a target that is worthwhile pushing people towards (as the endpoints will never be proved in our collective lifetime, says the pessimist).

Longest with _proven_ endpoints is worthwhile though.

As is either:
_Smallest_ with a gap >X for various Xs. Personally I'd like the Xs to start quite low, so that I can attack the problem with programming smarts (chinese remainder theorem plus sieves). This requires the maintainance of several tables though.
or:
_largest_ gap/endpoint ratio (=standard deviations from the mean). However, the 1132 (whatever) and the other small gaps dominate that table. So you'd need to bring in a cutoff somewhere, and again I think that 50000 is too high to interest me as a programming challenge.
(or both)

Phil

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