We show that for any polynomial f: Z Z with positive leading coefficient and irreducible over Q, if N is large enough then there are two strings of consecutive positive integers I₁=\n₁-m, , n₁+m\ and I₂=\n₂-m, , n₂+m\, where m = (N) (N) ^1/325525, such that I₁ I₂ 1, N, N = n₁ + n₂, and f (n) is composite for any n I₁ I₂. This extends the result in 5 which showed the same result but with f (n) =n.
Artyom Olegovich Radomskii (2025) studied this question.