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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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    No (non-constant) polynomial with integer coefficients yields prime numbers for all integer arguments.2012-12-07

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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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    @AmirHoseinSadeghiManesh: Like $y$our approach. +12012-12-15