Extremal Least Common Multiples in Rows of Pascal's Triangle

Abstract

For $r \geq 0$, 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: \[ a(n)=\min_{\substack{r\geq 0,\ S\subseteq\mathcal P_r\\ |S|=n}}\operatorname{lcm}(S). \] We first recast the fixed-row problem exactly as a weighted prime-power exclusion problem. This structural description explains why optimal supports may develop holes and yields an exact certification method for finite cases. Our main asymptotic result is \[ \log a(n)=2n+O\!\left(n\exp\!\left(-c\frac{(\log n)^{3/5}}{(\log\log n)^{1/5}}\right)\right) \] for some absolute $c>0$, so $a(n)^{1/n}\to e^2$. A two-band refinement further shows that every optimal row satisfies \[ r_n=2n+O\!\left(n\exp\!\left(-c\frac{(\log n)^{3/5}}{(\log\log n)^{1/5}}\right)\right), \] and that the minimum prefix defect of an optimal support is $o(n)$. Finally, two independent exact implementations certify the first finite structural transitions: $n=15$ is the first non-prefix optimum, while $n=41$ is the first case of minimum prefix defect greater than one.