Is there a generalization of Dirichlet's theorem along these lines?
If $p(n)$ is a polynomial of degree $k$ with positive integers as coefficients, such that the coefficients are relatively prime, then the sequence $p(1),p(2),p(3),\ldots$ has infinitely many primes?
$k=1$ is the good old Dirichlet's theorem on arithmetic progressions.
If the above is false, is there a trivial example for some degree '$k$'?
Best Answer
This is false as stated. Take, for example, $p(n) = n^2 + n$. The correct condition is that $p$ is irreducible and there exists no $d > 1$ such that $d | p(n)$ for all $n$. With this condition this is a big open problem, the Bunyakovsky conjecture, and it is open for any such polynomial $p$ of degree greater than $1$.