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Computing the Greatest Common Divisor of Binomial Coefficients binommnmk
Abstract
The greatest common divisor (GCD) of binom2n2k for 1≤ k<n is known to be some power of 2 times the product of all odd primes p such that 2n=pi+pj. We complete the analysis of this GCD by showing that this power of 2 is either 1 or 0 and relates it to Mersenne primes. We also show how to efficiently compute GCD {binommnmk: 1≤ k<n} when n and m satisfy certain conditions.
Versions
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BibTeX
@misc{wu2026computing,
title = {{Computing the Greatest Common Divisor of Binomial Coefficients $\binommnmk$}},
author = {Chai Wah Wu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.20940},
eprint = {2606.20940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.20940},
}
RIS
TY - UNPB
TI - Computing the Greatest Common Divisor of Binomial Coefficients binommnmk
AU - Chai Wah Wu
PY - 2026
DA - 2026/08/04
PB - arXiv
DO - 10.48550/arxiv.2606.20940
UR - https://arxiv.org/abs/2606.20940
AB - The greatest common divisor (GCD) of binom2n2k for 1≤ k<n is known to be some power of 2 times the product of all odd primes p such that 2n=p^i+p^j. We complete the analysis of this GCD by showing that this power of 2 is either 1 or 0 and relates it to Mersenne primes. We also show how to efficiently compute GCD {binommnmk: 1≤ k<n} when n and m satisfy certain conditions.
ER -