We show that for any polynomial f from the integers to the integers, with
positive leading coefficient and irreducible over the rationals, if x is
large enough then there is a string of (logx)(loglogx)1/835
consecutive integers nβ[1,x] for which f(n) is composite. This improves
a result of the first author, Konyagin, Maynard, Pomerance and Tao, which
states that there are such strings of length (logx)(loglogx)cfβ,
where cfβ depends on f and cfβ is exponentially small in the degree of
f for some polynomials.Comment: 22 page