[FOM] Prime values of polynomials

joeshipman@aol.com joeshipman at aol.com
Thu Mar 6 07:44:42 EST 2008

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.

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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