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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?

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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?

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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
@RossMillikan, you are correct. But I wondered as I saw his question when he was accent on without exhausting all the possibilities by trial and error ! But by what you said here I think he will earn better thought and will not be bothered on same case when numbers will be changed. – AmirHosein SadeghiManesh Dec 7 '12 at 7:11
@AmirHoseinSadeghiManesh: Like your approach. +1 – Babak S. Dec 15 '12 at 14:17

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