## Re: Is there another prime gap...

Expand Messages
• ... after ... all the primes below ... as I did in my ... divisible by 2, 3 or 5 and ... and that is as little as ... from a situation where ... Even with the
Message 1 of 6 , Feb 7, 2008
--- In primenumbers@yahoogroups.com, "julienbenney" <jpbenney@...> wrote:
>
> I have had a look at your factorisation of all the 1132 composites
after
> 1693182318746371 and it still does not seem as "efficient" in using
all the primes below
> the square root of that sixteen-digit number. Have you thought of,
as I did in my
> demonstration, of excluding all the numbers it that gap that are
divisible by 2, 3 or 5 and
> then comparing the factors?
>
> As it was I found 975 primes less than 975,000 occurring as factors,
and that is as little as
> 1.27 percent of all the primes over the duration. This is a far cry
from a situation where
> every possible prime less than the square root of a number occurs!
Even with the relativel
> persistent gap from 31397 to 31469 14 primes of 36 below the square
root of 31469 are
> required to divide all those numbers not divisible by 2, 3 or 5.
>
> I am really curious as to how much less probable it becomes that
every potential prime
> factor could occur over an unusually large prime gap. I do imagine
that there does exist a
> difficulty combining factors in a precise manner to create a large
prime gap, and that this
> difficulty increases immensely as one moves from four digits to
sixteen. Nonetheless, it is
> hard to believe this difficulty increases so much as to make even 1
percent of possible
> factors being used impossible with sixteen digits when all can be
used in a large four-digit
> gap is quite hard to believe.
>
> Is there any way of working how hard precisely combining factors to
create an unusually
> large gap is??

I'm supposing that there isn't such a way (now), or else we would have
a much better grip on how large gaps can be and where to find them.

Regarding the "efficiency" of prime factors in the gap, I noticed that
the gap of 13 composites between the primes 113 and 127 contains all
the primes up to and including 31.

The gap of 33 composites between the primes 1327 and 1361 contains all
the primes up to and including 43. This gap (I'm supposing) is the
last gap which contains all the primes up to the square root of where
the gap starts. So it *is* special. :)

Personally, I think the gap of 1131 composites between the primes
1693182318746371 and 1693182318747503 (which Jens mentioned) to be
the most outstanding, because it has the highest gap/(ln(p1))^2 ratio
(namely .9206).

The conventional "merit" scale of gap/ln(p1) seems to me to be more a
measure of the merit of those who are capable of searching high enough
to find such gaps, rather than of the gap itself! :)

Mark
Your message has been successfully submitted and would be delivered to recipients shortly.