Extremal Least Common Multiples in Rows of Pascal's Triangle

Abstract

For $r\ge 0$, let ${\mathcal{P}}_r=\{(\genfrac{}{}{0ex}{}{r}{0}),(\genfrac{}{}{0ex}{}{r}{1}),\dots ,(\genfrac{}{}{0ex}{}{r}{\lfloor r/2\rfloor })\}$ 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: $a(n)={\min }_{r\ge 0,S\subseteq {\mathcal{P}}_r,|S|=n}\mathrm{lcm}(S)$. 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 $\log a(n)=2n+O\left(n\exp \left(-c(\log n{)}^{3/5}(\log \log n{)}^{-1/5}\right)\right)$ for some absolute $c>0$, so $a(n{)}^{1/n}\to e^2$. A two-band refinement further shows that every optimal row satisfies $r_n=2n+O\left(n\exp \left(-c(\log n{)}^{3/5}(\log \log n{)}^{-1/5}\right)\right)$, and that the minimum prefix defect of an optimal support is $o(n)$.

arXiv comment18 pages, 3 figures. Revised exposition, streamlined proofs, and added figures. Ancillary files contain the reproducibility archive for the exact computations