On cubic polynomials giving many primes (Q919374)

The author finds 21 polynomials \(p=aT^ 3+bT^ 2+cT+d\in {\mathbb{Z}}[T]\) satisfying \(a=1\) or \(a=2\) and {\#}\(\{\) \(n\in {\mathbb{Z}}|\) \(0\leq n<100\); \(| p(n)|\) a prime number\(\}\geq 75\). For these polynomials p and for \(N=100\), 200, 300, 400 and 500 the numbers {\#}\(\{\) \(n\in {\mathbb{Z}}|\) \(0\leq n<N\); \(| p(n)|\) a prime number\(\}\) are given.