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What are some interesting sequences that contain infinitely many primes?

If it takes form of a polynomial, Dirichlet's theorem answer the question completely for linear polynomial. What about polynomials of degree more than 1? Is there a known polynomial of degree more than 1 that contains infinitely many primes?

What about more complicated sequences like $2^n+3^n$, $n!+1$, etc?

Please provide examples that are as interesting as possible, accompanied with proofs (or reference to proofs) if not too difficult.

Thanks in advanced.

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    http://math.stackexchange.com/q/77780/156602012-02-06
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    [Bunyakovsky conjecture](http://en.wikipedia.org/wiki/Bunyakovsky_conjecture)2012-02-06

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