# How do you know if a function only yields a certain kind of number?

Let's say that we have a function $3n^2-3n+13$. How do I know if the function only yields prime number without exhausting all the possibilities by trial and error?

-
It doesn't. Think of a clever choice of $n$ that kills primality. –  André Nicolas Dec 7 '12 at 6:26
No (non-constant) polynomial with integer coefficients yields prime numbers for all integer arguments. –  mjqxxxx Dec 7 '12 at 7:04

If your function is $f(n)=3n^{2}-3n+13$ by putting $n=13$ you will have $f(n)=13(39-3+1)$ that is not prime, even without thinking! why you ask it?
It is really $13(39-3+1)$ but the message is the same. Generally, for any polynomial you can set $n$ to any factor of the constant term. This takes care of all polynomials except those with a constant term of $\pm 1$ –  Ross Millikan Dec 7 '12 at 6:57