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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.2012-12-07
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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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