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Infinitely Many Binomial Coefficients of Deficiency One
Abstract
A binomial coefficient binomnk is good if all its prime divisors exceed k, and its deficiency is the number of k-smooth integers in (n−k,n]. A short constructive proof of the existence of infinitely many good binomial coefficients with deficiency exactly one is given, which provides an answer to a problem of Erdős, Lacampagne, and Selfridge.
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BibTeX
@misc{zhang2026infinitely,
title = {{Infinitely Many Binomial Coefficients of Deficiency One}},
author = {Xu Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25042},
eprint = {2609.25042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25042},
}
RIS
TY - UNPB
TI - Infinitely Many Binomial Coefficients of Deficiency One
AU - Xu Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.25042
UR - https://arxiv.org/abs/2609.25042
AB - A binomial coefficient binomnk is good if all its prime divisors exceed k, and its deficiency is the number of k-smooth integers in (n−k,n]. A short constructive proof of the existence of infinitely many good binomial coefficients with deficiency exactly one is given, which provides an answer to a problem of Erdős, Lacampagne, and Selfridge.
ER -