Re: [PrimeNumbers] Prime GAP of 364,188
- From: Jens Kruse Andersen
Date: 12/31/05 12:02:34
To: Prime Numbers
Subject: Re: [PrimeNumbers] Prime GAP of 364,188
Jose Ramón 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:
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.
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
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
[Non-text portions of this message have been removed]
- I wrote:
> 2) For each small prime, choose which numbers in the interval thatKermit 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 produceYes, 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