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Long intervals of consecutive composite values of polynomials
Abstract
Let f∈ℚ[t] be an irreducible integer-valued polynomial of positive degree with positive leading coefficient. We show that there is a constant cf>0 such that, for every sufficiently large real X, the interval (X/2,X] contains a block of at least cflog Xloglog X consecutive integers n for which f(n) is composite. The exponent of loglog X does not depend on the degree of f. This improves a result of Ford and Gabdullin, replacing a small fixed power of loglog X by loglog X. The result includes the case f(t)=t2+1.
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BibTeX
@misc{radomskii2026long,
title = {{Long intervals of consecutive composite values of polynomials}},
author = {Artyom Radomskii},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10762},
eprint = {2610.10762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10762},
}
RIS
TY - UNPB
TI - Long intervals of consecutive composite values of polynomials
AU - Artyom Radomskii
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.10762
UR - https://arxiv.org/abs/2610.10762
AB - Let f∈ℚ[t] be an irreducible integer-valued polynomial of positive degree with positive leading coefficient. We show that there is a constant c_f>0 such that, for every sufficiently large real X, the interval (X/2,X] contains a block of at least c_flog Xloglog X consecutive integers n for which f(n) is composite. The exponent of loglog X does not depend on the degree of f. This improves a result of Ford and Gabdullin, replacing a small fixed power of loglog X by loglog X. The result includes the case f(t)=t^2+1.
ER -