Does this polynomial produce only prime numbers?

  • #1
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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 anyone 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?
 
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Answers and Replies

  • #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.
 
  • #3

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