Long intervals of consecutive composite values of polynomials

Let f∈ Q[t] be an irreducible polynomial of positive degree with positive leading coefficient, taking integer values at every integer. We prove that there is a constant cf>0 such that, for every sufficiently large real X, the interval $(X/2,X]$ contains at least cflog Xloglog X consecutive integers n for which all values $f(n)$ are composite. The exponent of loglog X is independent of the degree of f. This improves the previous result of Ford and Gabdullin by replacing their small fixed power of loglog X with the full factor loglog X. In particular, the result applies to f(t)=t²+1.