- 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 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.

______________________________ Reply Separator _________________________________

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.. %.

Where do you keep this list?

Milton L. Brown

miltbrown@...

----- Original Message -----

From: <joe.mclean@...>

To: "Primes List" <PrimeNumbers@yahoogroups.com>

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.

>

>

> ______________________________ Reply Separator

_________________________________

> 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/

>

> - 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

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).

> these composite runs. Note that these are not primes gaps unless the

> endpoints are proved prime, as in the Dubner case.

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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