>

after

> I have had a look at your factorisation of all the 1132 composites

> 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?

and that is as little as

>

> As it was I found 975 primes less than 975,000 occurring as factors,

> 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.

every potential prime

>

> I am really curious as to how much less probable it becomes that

> 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.

create an unusually

>

> Is there any way of working how hard precisely combining factors to

> 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