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K-th Prime Proofs & Co-Prime Numbers

  1. Sep 23, 2010 #1
    I am having a hard time making head way on two problems related to the k-th prime and one about co-primes that I would really appreciate some help and/or direction!

    Prove that:
    (let pk be the k-th prime)
    Picture1-1.png

    and

    Picture3-2.png



    Regarding co-primes... is there any way to find a set of four numbers that are coprime, but cannot be subsequently grouped into sets of three that are?

    Again, thank you for your time and generosity in helping me out!
     
    Last edited: Sep 23, 2010
  2. jcsd
  3. Sep 23, 2010 #2
    In the first problem, is the number on the right prime or composite? If it's composite, what can you say about its factors?
     
  4. Sep 23, 2010 #3
    Oh, sorry!
    Uhmn... pk is the k-th prime, so pk + 1 (the right most term) would be a composite, I believe?
     
  5. Sep 23, 2010 #4
    I'm asking about the whole thing: 1 + the product of all primes up to pk.
     
  6. Sep 24, 2010 #5

    Gokul43201

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    I believe you are misreading the expression.

    To clarify: [itex]p_{k+1} \leq (p_1~p_2 \cdots p_k)+1[/itex]
     
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