Polynomials representing primes Suppose over $\mathbb{Z}$ we are given an irreducible polynomial $p(x)$.
Can we say that at the very least that $p(x)$ represents a prime as $x$ runs through integers?
Thanks in advance.
 A: Let $P(x)$ be a polynomial of degree $\ge 1$, with integer coefficients, such that no $d \gt 1$ divides all the coefficients. 
If $P(x)$ has degree $1$, then $P$ represents at least one prime. This is a consequence of Dirichlet's Theorem on primes in arithmetic progression (and easily implies that Theorem). 
As has been pointed out, for degree $\ge 2$, irreducibility is not enough to ensure that a polynomial represents a prime. For some irreducible polynomials $P(x)$, there exists a $d \gt 1$ such that $d$ divides $P(n)$ for every integer $n$. 
However, that can only happen for relatively simple congruential reasons. So let us focus attention on polynomials $P(x)$ for which there is no such universal $d$. Unfortunately, it is an open problem whether such a polynomial must necessarily represent at least one prime.  
Example: There is a good deal of evidence that there are infinitely many primes of the form $x^2+1$. However, whether or not there are infinitely many is a long-standing open problem, often called the Hardy-Littlewood Conjecture. If we could show that for all $a\ne 0$, (or even infinitely many $a$) there exists $x$ such that $(2ax)^2+1$ is prime, that would settle the Hardy-Littlewood Conjecture.  (Conversely, the Hardy-Littlewood Conjecture implies that there are infinitely many such $a$.) 
So the question you raised seems to be extremely difficult even for polynomials of degree $2$! 
A: No, e.g. irreducible $\rm\ f(x)\, =\, x(x+1)+4\ $ is even but $\rm\:f(x) \ne \pm 2.$
However, such fixed divisors of all values of $\rm\,f\,$ are essentially the only known obstruction to prime values. As motivation, let's start with a converse result. In $1918$ Stackel published the following simple observation:
Theorem  If $\rm\, f(x)\,$ is a composite integer coefficient polynomial then  $\rm\, f(n)\, $ is  composite for all $\rm\,|n| > B,\, $ for some bound $\rm\,B.\,$ In fact $\rm\, f(n)\, $ has at most $\rm\, 2d\, $ prime values, where  $\rm\, d = {\rm deg}(f)$.
The simple proof can be found online in Mott & Rose [3], p. 8.
I highly recommend this delightful and stimulating $27$ page paper
which discusses prime-producing polynomials and related topics.
Contrapositively, $\rm\, f(x)\, $ is prime (irreducible) if it assumes a prime value
for large enough $\rm\, |x|\, $. 
As an example, Polya-Szego popularized A. Cohn's irreduciblity test, which
states that $\rm\, f(x) \in \mathbb Z[x]\,$ is prime if  $\rm\, f(b)\, $
yields a prime in radix $\rm\,b\,$ representation (so necessarily $\rm\,0 \le f_i < b).$
For example $\rm\,f(x) = x^4 + 6\, x^2 + 1 \pmod p\,$ factors for all primes $\rm\,p,\,$
yet  $\rm\,f(x)\,$ is prime since  $\rm\,f(8) = 10601\rm$ octal $= 4481$ is prime.
Cohn's test fails if, in radix $\rm\,b,\,$ negative digits are allowed, e.g.
$\rm\,f(x)\, =\, x^3 - 9 x^2 + x-9\, =\, (x-9)\,(x^2 + 1)\,$ but $\rm\,f(10) = 101\,$ is prime.
Conversely Bouniakowski conjectured $(1857)$
that prime $\rm\, f(x)\, $ assume infinitely many prime values (excluding
cases where all the values of $\rm\,f\,$ have fixed common divisors, e.g. $\rm\, 2\: |\: x(x+1)+2\, ).$ However, except for linear polynomials (Dirichlet's theorem), this conjecture has never been proved  for any  polynomial of degree $> 1.$
Note that a result yielding the existence of one prime value extends to existence of infinitely many prime values, for any class of polynomials closed under shifts, viz. if $\rm\:f(n_1)\:$ is prime, then $\rm\:g(x) = f(x+ n_1\!+1)\:$ is prime for some $\rm\:x = n_2\in\Bbb N,\:$ etc.
For further detailed discussion of Bouniakowski's conjecture and related results, including heuristic and probabilistic arguments, see Chapter 6 of Ribenboim's The New Book of Prime Number Records. 
[1] Bill Dubuque, sci.math 2002-11-12, On prime producing polynomials.
[2] Murty, Ram. Prime numbers and irreducible polynomials.
Amer. Math. Monthly, Vol. 109 (2002), no. 5, 452-458.  
[3] Mott, Joe L.; Rose, Kermit. Prime producing cubic polynomials.
Ideal theoretic methods in commutative algebra, 281-317.
Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001.  
