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Math Help - Composite Numbers

  1. #1
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    Composite Numbers

    Show that if k is any positive integer, then the numbers (k+1)! + j, where j = 2,3,...,k+1, are all composite.

    I believe this is essentially saying that there exists arbitrarily long sequences of consecutive composite numbers. I am just having a little trouble with the proof. Any suggestions would be great.

    Thanks!
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  2. #2
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    Quote Originally Posted by JoAdams5000 View Post
    Show that if k is any positive integer, then the numbers (k+1)! + j, where j = 2,3,...,k+1, are all composite.

    I believe this is essentially saying that there exists arbitrarily long sequences of consecutive composite numbers. I am just having a little trouble with the proof. Any suggestions would be great.

    Thanks!

    Simply noting that
    j|[(k+1)! + j] , for all j = 2,3,...,k+1
    should do I guess.

    Or I am not getting it?
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  3. #3
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    Would this be a good argument?

    Given that (k + 1)! = 1(2)(3)(4)...(k - 1)k(k + 1), it's evident that every positive integer is covered in that factorial up through k + 1. So no matter what j is, you can evenly divide j into (k + 1)! (and obviously j evenly divides j) leaving (k + 1)!/j + 1 from the given problem.
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