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Does this polynomial produce only prime numbers?

  1. Nov 17, 2011 #1
    I'm trying to find a general formula for an algebraic equation, I'm studying the behavior of [itex]∏_{i=2}^n(1-\frac{1}{i^m})[/itex] for m=3 and so far I've seen that I can find a general formula if n^2 + n + 1 produces only prime numbers. if not, it would get way harder to find a general formula for it by my intuitive method. Does any one have any ideas? Do you know a counter-example? I've tried n=1,2,3,4,5,6 and it seems fine. I think a counter-example wouldn't be so easy to find. anyone can help?
    For the experts on this forum who are into number theory, is there a polynomial of any degree that produces only prime numbers?
     
    Last edited: Nov 17, 2011
  2. jcsd
  3. Nov 17, 2011 #2

    micromass

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    It doesn't work for 7:

    [tex]7^2+7+1=57=3\cdot 19[/tex]

    I don't think a polynomial exists which only generates primes.
     
  4. Nov 17, 2011 #3

    micromass

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