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Prime Number Gaps

  1. Jul 10, 2006 #1
    Hello everyone,

    I'd first like to say that I am uninformed on this subject and that I have a question to the mathematicians on these forums who know about the subject.

    In the set of all prime numbers, has the integer gaps between two prime numbers been studied? I mean, do mathematicans know what the largest difference is between two prime numbers?

    Im interested in this subject and I would like to know..

    Thanks in advance.

    --rad
     
  2. jcsd
  3. Jul 10, 2006 #2

    shmoe

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    The gap can be arbitrarily large. Just consider n!+2, n!+3,...n!+n.

    Lots of work has been done, you might want to take a look at Caldwell's prime pages: http://primes.utm.edu/notes/gaps.html
     
  4. Aug 5, 2006 #3

    CRGreathouse

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    Since the primes have measure 0, prime gaps must be unbounded in length. Think about it -- if every k integers had a prime number for some fixed k, then some primes would have common (nontrivial) factors.
     
  5. Aug 5, 2006 #4

    mathwonk

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    i refer you to the work of helmut maier.

    Helmut Maier
    Primes in short intervals.

    Source: Michigan Math. J. 32, iss. 2 (1985), 221
     
  6. Aug 7, 2006 #5

    matt grime

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    the integers have measure zero, but the gaps between consecutive integers is not unbounded.
     
  7. Aug 8, 2006 #6

    CRGreathouse

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    What I mean is that for a set [tex]X\in\mathbb{N}[/tex] with

    [tex]\lim_{n\rightarrow\infty}\frac1n\sum_{x\in X|x\le n}x=0[/tex]

    [tex]\forall n\in\mathbb{N}\;\;\exists m\in\mathbb{N}[/tex] such that there is no [tex]x\in X[/tex] with [tex]m\le x\le m+n[/tex]. (The set of primes is of course such a set by the PNT.) I'm sorry if I was ambiguous.
     
    Last edited: Aug 8, 2006
  8. Aug 8, 2006 #7

    shmoe

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    You want:

    [tex]\lim_{n\rightarrow\infty}\frac1n\sum_{x\in X|x\le n}1=0[/tex]

    or equivalently here pi(n)/n->0 as n->infinity. In otherwords, the asymptotic density of the primes is zero.
     
  9. Aug 8, 2006 #8
    According to Bertrand's postulate, there is at least one prime between n and 2n-2, for any n>3. I wonder if there is a theorem about the number of primes between n and 2n exclusive (see http://www.research.att.com/~njas/sequences/A060715 ), because that number seems to be steadily increasing over a sufficiently large period of the sequence (sorry if this is not precise enough); what I mean is that, for n=5, for instance, the number of primes between n and 2n is 1, but, it seems, for any n > 5 the number of primes between n and 2n is greater than 1; similarly, for n= 8, the number of primes between n and 2n is 2, but for any n>8, the number of primes between n and 2n is greater than 2 (?), and so on.

    If it is true that for any m >= 1 there is an n for which the number of primes between k and 2k, k>=n, is greater than m (is it?) then there is a limit on prime gaps as well, depending on m (or n), I think (although Bertrand's postulate itself puts a limit on prime gaps, depending on n).

    What I mean by Bertrand's postulate putting a limit on prime gaps is that for any prime p, there is another prime between p+1 and 2p.
     
    Last edited: Aug 8, 2006
  10. Aug 8, 2006 #9

    shmoe

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    pi(2n)-pi(n)~n/log(n) by the prime number theorem, so the number of primes in [n,2n] tends to infinity as n does.

    The bound Bertrands puts on the gap to the next prime is pretty far from what's known to be true (though correspondingly simpler to prove!). For example, if n is large enough, we can guarantee a prime in [n,n+n^0.525].
     
  11. Aug 8, 2006 #10

    matt grime

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    density, not measure.
     
  12. Aug 9, 2006 #11

    CRGreathouse

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    Oops, you're absolutely right. That's what I meant.
     
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