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.
For integers $m\geq 2$ and $m\mid N$, let $G(N;m)=\gcd\{\binom{N}{k}:0<k<N,\ m\mid k\}$. We prove a complete $p$-adic valuation formula for $G(N;m)$ at primes $p\equiv -1\pmod m$, under the hypotheses $m\geq 3$, $m\mid N$, and $m<N$. Writing $N=\sum_i d_i p^i$, put $A=\sum_{i\text{ even}}d_i$ and $B=\sum_{i\text{ odd}}...
For $k\geq2$, let $\alpha_k$ be the supremum of the exponents $a$ for which almost every integer $n$ has $k$ distinct divisors in a multiplicative interval of relative length $(\log n)^{-a}$. Select each positive integer $i$ independently with probability $1/i$, forming a random set $\mathbf A$, and let $\beta_k$ be th...
Let $p_n$ be the $n$th prime and $g_n=p_{n+1}-p_n$. Following Gafni and Tao, an integer $m$ with $p_n<m<p_{n+1}$ is rough for the gap if its least prime factor is at least $g_n$; let $M_n$ be the number of such integers. Gafni and Tao proved that the number of $n$ with $X\le p_n\le 2X$ and $M_n=0$ is $O(X/\log^2 X)$, a...
For an integer $n>1$, let $P^+(n)$ be the largest prime factor of $n$, and let $P_y^+(n)$ denote the largest prime factor of $n$ not exceeding $y$. One of Erd\H{o}s and Tur\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is $1/2$. Furthermore, Wang conjectures that for...
For $f(X)=X^r(X^s+a)\in\mathbb F_{p^m}[X]$, with $m,r,s\geqslant1$ and $a\in\mathbb F_{p^m}^\times$, put $g=\gcd(s,p^m-1)$. We prove that, for each fixed $g$, only finitely many prime characteristics $p$ admit such permutation binomials, even as $m,r,s,a$ vary. If $6\nmid g$, every such $p$ divides Wendt's determinant...
For an integer $k\geq2$, let $\alpha_k$ be the supremum of the real numbers $a$ for which almost every integer $n\geq2$ has divisors $d_1<\cdots<d_k\mid n$ satisfying $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr).$ Let $\mathcal A\subseteq\N$ be the logarithmic random set in which the events $m\in\mathcal A$ are mutually ind...
Ya-Ping Mao, Yan-Yan Song· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
