Irreducible Values of Polynomials: a Non-analogy

For a polynomial f(T ) ∈ Z[T ], the frequency with which the values f(n) are prime has been considered since at least the 18-th century. Euler observed, in a letter to Goldbach in 1752, that the sequence n + 1 has many prime values for 1 ≤ n ≤ 1500. Legendre assumed an arithmetic progression an+ b with (a, b) = 1 contains infinitely many primes in his work on the quadratic reciprocity law. There is also the old question of twin prime pairs n and n+2, but we will focus here only on a single polynomial (in one variable). An asymptotic estimate for