Long strings of consecutive composite values of polynomials

Ford, Kevin; Gabdullin, Mikhail

doi:10.1090/tran/9348

Long strings of consecutive composite values of polynomials
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by Kevin Ford and Mikhail R. Gabdullin;

Trans. Amer. Math. Soc. 378 (2025), 1261-1282

DOI: https://doi.org/10.1090/tran/9348

Published electronically: December 30, 2024

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Abstract:

We show that for any polynomial $f:\mathbb {Z}\to \mathbb {Z}$ with positive leading coefficient and irreducible over $\mathbb {Q}$, if $x$ is large enough then there is a string of $(\log x)(\log \log x)^{1/835}$ consecutive integers $n\in [1,x]$ 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 $f$.

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