A solution to the Erdős-Sárközy product-density problem
Abstract
Color each positive integer red or blue, and call an integer a same-color product if it is the product of two red numbers or of two blue numbers. Does there exist a universal constant c > 0 such that: For each fixed coloring, and for all sufficiently large M (where "sufficiently large" may depend on the coloring), at least cM of the integers among $1$, $2$, …, M are same-colored products (allowing for equal factors, and where each distinct product is counted once)? This problem was given (in an equivalent way) in Guy's famous text on unsolved problems in number theory, and was posed (in equivalent ways) by Erdős and Sárközy in 1990, in the proceedings of the 1988 Banff number theory conference, and 1993 [Colloq. Math.], and this problem seems to have remained open. We succeed in solving this problem, by proving that no such universal constant exists.
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BibTeX
@misc{campbell2026solution,
title = {{A solution to the Erdős-Sárközy product-density problem}},
author = {John M. Campbell},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01480},
eprint = {2611.01480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01480},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A solution to the Erdős-Sárközy product-density problem
AU - John M. Campbell
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01480
UR - https://arxiv.org/abs/2611.01480
AB - Color each positive integer red or blue, and call an integer a same-color product if it is the product of two red numbers or of two blue numbers. Does there exist a universal constant c > 0 such that: For each fixed coloring, and for all sufficiently large M (where "sufficiently large" may depend on the coloring), at least cM of the integers among $1$, $2$, …, M are same-colored products (allowing for equal factors, and where each distinct product is counted once)? This problem was given (in an equivalent way) in Guy's famous text on unsolved problems in number theory, and was posed (in equivalent ways) by Erdős and Sárközy in 1990, in the proceedings of the 1988 Banff number theory conference, and 1993 [Colloq. Math.], and this problem seems to have remained open. We succeed in solving this problem, by proving that no such universal constant exists.
ER -