Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook