AI 中文总结
本文证明受限二项式GCD的p-adic极值恰为r_p(m)，并构造达到该值的行，对m=p^a+1精确给出最小极值行p^{3a}+1，结合Kummer定理与显式见证。
AI 中文摘要
对于整数 $m\ge 2$ 以及满足 $N>m$ 且 $m\mid N$ 的行 $N$，考虑受限二项式最大公约数 $G(N;m)=\gcd\{\binom Nk:0<k<N,\\ m\mid k\}$。固定一个素数 $p$ 满足 $p\nmid m$，并令 $r_p(m)$ 为使得 $m<p^r$ 的最小正整数 $r$。我们证明，当 $N$ 遍历所有允许的行时，$v_p(G(N;m))$ 的最大可能值恰好等于 $r_p(m)$，并给出一个构造性的等式行。在最小极值行 $T_p(m)$ 被定义之前，该达到性已被确立。对于特殊族 $m=p^a+1$（其中 $a\ge2$），我们精确确定了该最小行：$T_p(p^a+1)=p^{3a}+1$。证明结合了 Kummer 进位定理与一个用于普遍上界的显式前导数字见证、一个用于等式的乘法阶构造，以及一个在 $p^{3a}+1$ 之下独立的严格最小性论证，其中包含一个用于唯一最坏前导数字模式的回退见证。所选 GCD 族本身在文献中已知；与早期结果的关系以及文献检索的局限性被明确陈述。
英文摘要
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'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.
Comments13 pages, no figures
