# Does this polynomial produce only prime numbers?

I'm trying to find a general formula for an algebraic equation, I'm studying the behavior of $∏_{i=2}^n(1-\frac{1}{i^m})$ 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:

micromass
Staff Emeritus
Homework Helper
It doesn't work for 7:

$$7^2+7+1=57=3\cdot 19$$

I don't think a polynomial exists which only generates primes.

micromass
Staff Emeritus