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Prove that none of them is prime

  1. Mar 18, 2009 #1
    1. The problem statement, all variables and given/known data

    Let n be a part of the natural numbers, with n>=2. Consider the numbers [n factorial +2 ], [n factorial + 3], ..., [n factorial + n].
    Prove that none of them is prime, and deduce that there are arbitrarily long finite stretches of consecutive non-prime nummbers in the natural numbers.


    2. Relevant equations



    3. The attempt at a solution

    I really don't know how to do this question. Any help/hints would be appreciated.
     
  2. jcsd
  3. Mar 18, 2009 #2

    tiny-tim

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    Hi kmeado07! :smile:

    Hint: what factor is there of (n! + 3)? :wink:
     
  4. Mar 18, 2009 #3
    Re: Primes

    Is (n factorial + 2) a factor of it?

    Which would then apply to the others, which would show that none of them were prime.
    For them each to be prime, their only factors would have to be themselves and 1.
     
  5. Mar 18, 2009 #4

    tiny-tim

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    Hi kmeado07! :smile:

    (please use the !)
    No: n! + 2 is only one less than n! + 3 …

    how can it be a factor of it?

    Try writing n! + 3 in full (with n = 7, say) :smile:
     
  6. Mar 18, 2009 #5
    Re: Primes

    ok, so a factor of n!+3 is 3, and a factor of n!+2 is 2 and so on.
    So a factor of n!+n is n. This shows that none of them is prime.

    How would i go about doing the second part of the question?
     
  7. Mar 18, 2009 #6

    tiny-tim

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    ok, so how many consecutive non-prime numbers are there starting with n!?
     
  8. Mar 18, 2009 #7
    Re: Primes

    So there are n-2 consecutive non-primes starting with n! ?
     
  9. Mar 18, 2009 #8

    tiny-tim

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    ooh, i got that slightly wrong, didn't i? :redface:

    yes, n-1 starting with n! + 2 …

    so if you wanted 1,000,000 consecutive non-primes, where would you start? :smile:
     
  10. Mar 18, 2009 #9
    Re: Primes

    you would start at n! + 1,000,003 ?
     
  11. Mar 18, 2009 #10

    tiny-tim

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    Nowhere near …

    try again :smile:
     
  12. Mar 18, 2009 #11
    Re: Primes

    i don't know, im confused!
     
  13. Mar 18, 2009 #12
    Re: Primes

    bump i'd like more info on this too
     
  14. Mar 18, 2009 #13

    HallsofIvy

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    Re: Primes

    If n- 2= 1,000,000, what is n? It's that easy.
     
  15. Mar 18, 2009 #14
    Re: Primes

    I really have no idea whats going on starting with this

    are you saying for n=10, then

    2 + 10*9*8*...*2*1

    has n-1 = 9 consectuve non primes?

    because I dont see that, maybe im not understanding the question..
     
  16. Mar 18, 2009 #15

    D H

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    Re: Primes

    Which numbers from 2 to 10 fails to divide 10! ? Given that, what divides 10! + 2? 10! + 3, and so on?
     
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