# [FOM] Prime values of polynomials

Grant Olney Passmore grant at math.utexas.edu
Thu Mar 6 14:21:58 EST 2008

```joeshipman at aol.com wrote:
> It's not really tougher; if f(x) has only finitely many non-prime
> values, let the largest x for which f(x) is composite be k; then f(x^2
> + k + 1) has only prime values.

Wonderfully slick!  How right you are.  I had a much more difficult
proof in mind :).  Thanks for pointing this out.

Grant

>
> Similarly, any polynomial with only finitely many prime values can be
> used to make a polynomial with no prime values. But it's very striking
> to me that we can't prove any polynomial of degree >1 has infinitely
> many prime values, even for polynomials of the form x^2 + k for k>0.
>
> -- JS
>
> -----Original Message-----
> From: Grant Olney Passmore <grant at math.utexas.edu>
>
>
> Theorem: No *univariate* polynomial (in Q[x]) of degree greater than 0
> exists that takes on prime values for all but finitely many integer
> values of its variable.
>
> This one is tougher.
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```