Progressions With Squares In an arithmetic progression like $3+5n$, what happens when we square $n$ and rewrite the progression as $3+(5 n^2)$? It is no longer arithmetic, right? What do we call this kind of progression? Does it have infinitely many primes? Can Dirichlet's theorem be used?
 A: I don't know if it has a standard name. "Quadratic progression" seems reasonable to me.  Dirichlet's Theorem does not apply.  Whether such progressions have infinitely many primes is unknown.  Even the case of whether there are infinitely many primes of the form $n^2+1$ is unknown.  (Of course in some cases you can rule out infinitely many primes because of a common factor in the coefficients, or the quadratic being factorable.  For instance $n^2-9=(n-3)(n+3)$ certainly doesn't have infinitely many primes in its progression.)
A: We can call it a quadratic sequence, though I would prefer to use quadratic polynomial. 
For some quadratic polynomials $P(x)$ with integer coefficients, it is easy to find a $d\gt 0$ such that $d$ divides $P(x)$ for every integer $x$. 
Apart from such cases, the problem of whether a quadratic polynomial represents infinitely many primes is open. It has been conjectured that except in "obvious" cases, $P(x)$ always does represent infinitely many primes. There are even conjectured analogues of the Prime Number Theorem, which seem to fit the facts quite well.  
But there is no quadratic polynomial $P(x)$ which has been proved to represent infinitely many primes. In the other direction, apart from trivial cases, there is no quadratic polynomial with integer coefficients which has been proved to represent only finitely many primes.
The most famous open problem of this type is whether the polynomial $x^2+1$ represents infinitely many primes. 
For a more general conjecture of this type, please see the Bunyakovsky Conjecture.
