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Re: Is there another prime gap...

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  • Mark Underwood
    ... 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
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      --- 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
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