LONG STRINGS OF CONSECUTIVE COMPOSITE VALUES OF POLYNOMIALS

We show that for any polynomial f : Z → Z with positive leading coefficient and irreducible over Q, if x is large enough then there is a string of (log x)(log log x)^1/835 consecutive integers n ε [1, x] for which f(n) is composite. This improves the result by Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, and Terence Tao [J. Eur. Math. Soc. (JEMS) 23 (2023), pp. 667–700], which has the exponent of log log x being a constant depending on f which can be exponentially small in the degree of f.