I am trying to make inferences about quadratic forms $f(n) = x^2+ax+b$ that produce primes for some integers $0,1,2,...$ (upto a finite limit). I noted that $f(0)$ must be prime so $b$ must be prime. Also is it necessary that $a$ should be prime or something like this $$(|a| \le 1) \vee (|b|\le 1)\vee(\text{prime}(|a|)\wedge \text{prime}(|b|))$$ What can we further deduce?