Extremal Least Common Multiples in Rows of Pascal's Triangle
Abstract
For r ≥ 0, let 𝒫r={binomr0,binomr1,…,binomr⌊ r/2⌋} be the set of distinct entries in row r of Pascal's triangle. We study the least possible least common multiple of n entries chosen from one row, with the row itself also free: a(n)=minr≥0, S⊆𝒫r, |S|=nlcm(S). Whereas the least common multiple of an entire row is given by a classical identity of Farhi, allowing both the row and the selected coefficients to vary creates a different optimization problem. We first recast the fixed-row problem exactly as a weighted prime-power exclusion problem, which explains how omitting a few coefficients can remove expensive prime-power contributions. Our main asymptotic result is log a(n)=2n+O (nexp (−c(log n)3/5(loglog n)−1/5)) for some absolute c>0, so a(n)1/n→ e2. A two-band refinement further shows that every optimal row satisfies rn=2n+O (nexp (−c(log n)3/5(loglog n)−1/5)), and that the minimum prefix defect of an optimal support is o(n).
Related
- Least common multiples of polynomial values: averaging, ray classes, and Galois statistics
- Pascal tiling and congruences modulo N in Pascal's triangle
- A sharp bound for the least signless Laplacian eigenvalue of triangle-free graphs
- Sharp p-adic Extrema and Least Extremal Rows for Restricted Binomial GCDs
- Pascal-like Sprugnoli arrays
BibTeX
@misc{huber2026extremal,
title = {{Extremal Least Common Multiples in Rows of Pascal's Triangle}},
author = {Felix Huber},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.23024},
eprint = {2609.23024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23024},
}
RIS
TY - UNPB
TI - Extremal Least Common Multiples in Rows of Pascal's Triangle
AU - Felix Huber
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.23024
UR - https://arxiv.org/abs/2609.23024
AB - For r ≥ 0, let 𝒫_r={binomr0,binomr1,…,binomr⌊ r/2⌋} be the set of distinct entries in row r of Pascal's triangle. We study the least possible least common multiple of n entries chosen from one row, with the row itself also free: a(n)=min_r≥0, S⊆𝒫_r, |S|=nlcm(S). Whereas the least common multiple of an entire row is given by a classical identity of Farhi, allowing both the row and the selected coefficients to vary creates a different optimization problem. We first recast the fixed-row problem exactly as a weighted prime-power exclusion problem, which explains how omitting a few coefficients can remove expensive prime-power contributions. Our main asymptotic result is log a(n)=2n+O (nexp (−c(log n)^3/5(loglog n)^−1/5)) for some absolute c>0, so a(n)^1/n→ e^2. A two-band refinement further shows that every optimal row satisfies r_n=2n+O (nexp (−c(log n)^3/5(loglog n)^−1/5)), and that the minimum prefix defect of an optimal support is o(n).
ER -