- From: Jens Kruse Andersen

Date: 12/31/05 12:02:34

To: Prime Numbers

Subject: Re: [PrimeNumbers] Prime GAP of 364,188

Jose RamÃ³n Brox wrote:

> > There are better methods to choose which numbers have small factors.

AFAIK, all efficient searches for large gaps between large primes/prp's (e.g

> > Such methods were used for the 2 known Megagaps:

> > http://hjem.get2net.dk/jka/math/primegaps/megagap.htm

>

> Indeed a very intelligent approach: I was wondering how all that records

> could be arrived at, and now I know. Are there any other approaches,

> or yours is the main used one?

above 50 digits) have included some variant of these points:

********

From Kermit

kermit@...

Has anyone tried this approach?

To product a prime gap

choose prime divisors,

p1, p2, p3, .... pN.

Lets pick p1 = 2, p2 = 3, p3 = 5, etc.

Choose k = -1 mod 2. This makes k odd.

choose k = -2 mod 3.

Now we don't need to worry about -3, -5, -7, etc because these are all taken

care of by

choosing k = -1 mod 2.

choose k = - 4 mod 5.

not we don't need to worry about -5, - 8, - 11, - 14, etc, because these are

already taken care of by k = -2 mod 3.

choose k = -6 mod 7.

Now we don't need to worry about -9, -14, -19, - 24, etc because these are

taken care of

by k = -4 mod 5.

Choose k = - 10 mod 11

Now we don't need to worry about -6, -13, -20, etc because these are taken

care of by

k = -6 mod 7.

etc.

So with the 5 primes, 2,3,5,7,11 we can assure a prime gape of length 11.

Each prime we add to the list lengthen's the prime gap by more than 1.

In fact, note that we just defined k = 1 mod 2, k=1 mod 3, k = 1 mod 5, k=1

mod 7,

So the minimum k satisfying this is

k = 2 * 3 * 5 * 7 * 11 + 1 = 2311.

In fact, I just realized why the pattern I chose will always yield p#+1 as

the value of

k. Thus I just proved that the prime gap of p# is at least p.

Perhaps some other way of assigning the negative values will produce a sieve

of higher

merit.

[Non-text portions of this message have been removed] - I wrote:
> 2) For each small prime, choose which numbers in the interval that

Kermit Rose wrote:

> prime should divide, trying to minimize the "overlap" with other primes.

> Has anyone tried this approach?

.....

Yes, that is the primorial approach mentioned several times in this thread.

> Perhaps some other way of assigning the negative values will produce

Yes, that is exactly what the choices in 2) is about.

> a sieve of higher merit.

Maybe you didn't have time to read when you posted 6 mails in 2h 12min...

--

Jens Kruse Andersen