Extremal Least Common Multiples in Rows of Pascal's Triangle

Abstract

For $r \ge 0$, let $\mathcal P_r=\{\binom{r}{0},\binom{r}{1},\ldots,\binom{r}{\lfloor r/2\rfloor}\}$ 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: $a(n)=\min_{r\ge0,\ S\subseteq\mathcal P_r,\ |S|=n}\operatorname{lcm}(S)$. 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 $o(n)$.