Sharp $p$-adic Extrema and Least Extremal Rows for Restricted Binomial GCDs

Abstract

For integers $m\ge 2$ and rows $N>m$ divisible by $m$, consider the restricted binomial greatest common divisor $G(N;m)=\gcd\{\binom Nk:0<k<N,\ m\mid k\}$. Fix a prime $p$ with $p\nmid m$, and let $r_p(m)$ be the least positive integer $r$ such that $m<p^r$. We prove that the largest possible value of $v_p(G(N;m))$, as $N$ ranges over all admissible rows, is exactly $r_p(m)$, and we give a constructive equality row. Attainment is established before the least extremal row $T_p(m)$ is defined. For the special family $m=p^a+1$ with $a\ge2$, we determine that least row exactly: $T_p(p^a+1)=p^{3a}+1$. The proof combines Kummer&#39;s carry theorem with an explicit leading-digit witness for the universal upper bound, a multiplicative-order construction for equality, and a separate strict-minimality argument below $p^{3a}+1$ with a fallback witness for the unique worst leading-digit pattern. The selected-GCD family itself is known in the literature; the relation to earlier results and the limits of the documented literature search are stated explicitly.