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Gaps between primes

  1. Jun 14, 2012 #1
    is there any formula to compute the gaps between primes which could be true to all prime numbers?..thanks..please help!
     
  2. jcsd
  3. Jun 14, 2012 #2
    we don't have a formula to generate prime numbers , research is going on , if we have such a formula that will also give the gap between two successive primes
     
  4. Jun 14, 2012 #3

    Mentallic

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    Knowing this would make you very rich, wouldn't it?
     
  5. Jun 14, 2012 #4

    phinds

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    How?
     
  6. Jun 14, 2012 #5

    Mentallic

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    I thought I read about it somewhere, but here's a link to such a claim that money is involved with finding large primes:

    https://www.eff.org/awards/coop

    If there exist formulae to calculate the gaps between primes, then surely they'd be able to find a lot more primes than just searching for all the Mersenne primes.
     
  7. Jun 14, 2012 #6

    HallsofIvy

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    There exist some monetary prizes for math papers but none of them would make you rich!
     
  8. Jun 14, 2012 #7

    coolul007

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    One can create arbitrary large consecutive composite integers by the sequence:
    (k+1)! + 2, (k+1)! + 3, ...,(k+1)! + k, (k+1)! + k + 1
    This sequence gives you k consecutive integers that are not prime
     
  9. Jun 14, 2012 #8

    Mentallic

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    Well that's the first time I've seen anyone downsize the value of hundreds of thousands of dollars (millions if you include the Millenium prizes).
     
  10. Jun 14, 2012 #9
    . .Thank y0u guys f0r your kind replies. . .I just need s0me ideas to put on with my research paper. .Thanks for sharing, it would be a great help.
     
  11. Jun 15, 2012 #10

    haruspex

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    Try oogling Bertrand's postulate, twin prime conjecture, prime number theorem.
     
  12. Jun 15, 2012 #11
    Can you prove that there are arbitrarily large gaps between consecutive primes? In other words there's a gap of a million, a gap of a billion, a gap of a zillion ... you can make the gap between consecutive primes as large as you want. It's an elementary proof, no advanced math needed.
     
  13. Jun 15, 2012 #12

    coolul007

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    The gaps are not necessarily between consecutive primes, they are arbitrarily large consecutive composites.
     
  14. Jun 15, 2012 #13
    Oh boy..the minute I saw this post I thought:
    Given two consecutive primes p1 and p2 the gap between then is |p1 - p2|.
    :-D
     
  15. Jun 15, 2012 #14
    Sorry, of course that's what I meant :-)

    ps -- I see you mentioned this earlier.
     
    Last edited: Jun 15, 2012
  16. Jun 16, 2012 #15
    There should be a pattern. Primes are not multiples of 2, not multiples of 3, not multiples of 4, etc. Just take the numbers that are not multiples of anything
     
  17. Jun 16, 2012 #16
    Its easy to say "there should be". Try finding it!
     
  18. Jun 16, 2012 #17
    So...all we have to do is just take all of the prime numbers? Great.

    Your statement isn't even true. Most prime numbers are not multiples of three...
     
  19. Jun 16, 2012 #18
    You're right, but that doesn't give any new information.

    We know 2 is prime and no other prime is divisible by 2.

    We know 3 is prime and no other prime is divisible by 3.

    We don't need to consider 4 because if a number is divisible by 4, it's already divisible by 2, which we checked earlier.

    We know 5 is prime and no other prime is divisible by 5.

    Continuing like this, we see that we could figure out the distribution of primes ... if we already knew the distribution of primes. We haven't gotten any more insight.

    However, your idea is actually the basis of the famous Sieve of Eratosthenes. You start with a list of all the counting numbers from 2 onward. You draw a circle around two; then you cross out 4,6,8, and all the other multiples of 2.

    Then you put a circle around 3; and cross out all the multiples of 3. Continuing like this, you end up with all the primes circled. You can use this algorithm to find all the primes below any given number. The algorithm's about 2300 years old -- and still as good as ever.

    http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

    There's a very cool animation on that page showing the algorithm in action.
     
  20. Jun 16, 2012 #19

    coolul007

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    It's akin to proving a negative proposition. "It's not one of these!" That applies to a lot of things. The closest we will come is proving the Goldbach Conjecture, that places two primes equidistant from a fixed point.
     
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