Xinyu Sun, Victor H. Moll
The p-adic valuations of a sequence of integers T(n) counting alternating sign matrices is examined for p=2 and p=3. Symmetry properties of their graphs produce a new proof of the result that characterizes the indices for which T(n) is odd.
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We present a binary tree that describes the 2-adic valuation of a sequence of coefficients arising from the evaluation of a rational integral.
We analyze properties of the 2-adic valuations of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence. Connections with the orbits arising from the Collatz (3x+1) problem are discussed.
The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties.
The 2-adic valuation of the general degree 2 polynomial in 2 variables
本文研究了多项式的p-adic赋值问题，通过赋值树方法描述了2变量一般2次多项式的2-adic赋值树。
We analyze congruence classes of $S(n,k)$, the Stirling numbers of the second kind, modulo powers of 2. This analysis provides insight into a conjecture posed by Amdeberhan, Manna and Moll, which those authors established for $k\le5$. We provide a framework that can be used to justify the conjecture by computational means, which we then complete for $k=5, 6,..., 20$.
The 2-adic valuation of the Stirling numbres is examined. We conjecture pattrens about the distributions of these valuations in residue classes modulo powers of 2.
In 1992 Strauss, Shallit and Zagier proved that for any positive integer $a$ we have $$\sum_{k=0}^{3^a-1}\binom{2k}{k}=0 (mod 3^{2a})$$ and furthermore $$3^{-2a}}\sum_{k=0}^{3^a-1}\binom{2k}k=1 (mod 3).$$ Recently a $q$-analogue of the former congruence was conjectured by Guo and Zeng. In this paper we prove the Guo-Zeng conjecture and also give a $q$-analogue of the latter congruence.
Some restricted partition functions in terms of 2-adic valuation
本文研究受限分拆函数的表示问题，利用整数的2进赋值给出了若干此类函数的表达形式。
The aim of this paper is to prove conjectures concerning $p$-adic valuations of Stirling numbers of the second kind $S(n,k)$, $n,k\in\mathbb{N}_+$, stated by Amdeberhan, Manna and Moll and Berrizbeitia et al., where $p$ is a prime number. The proof is based on elementary facts from $p$-adic analysis.
Universal Matrices for Counting Fibonomial and $C$-nomial Coefficients by their $p$-adic Valuations
针对由序列 $C$ 定义的二项式系数推广形式 $C$-nomial 系数，提出当 $C$ 为强整除序列时，其按 $p$-adic 赋值计数的生成函数可通过通用矩阵乘积公式表示，并将该公式推广至 $C$-多项式系数。
On the 3-adic Valuation of a Cubic Binomial Sum
本文证明了关于立方二项式和的3-进制估值的猜想，提供了该领域的新的数学结果。
We revisit Huber's theory of continuous valuations, which give rise to the adic spectra used in his theory of adic spaces. We instead consider valuations which have been reified, i.e., whose value groups have been forced to contain the real numbers. This yields reified adic spectra which provide a framework for an analogue of Huber's theory compatible with Berkovich's construction of nonarchimedean analytic spaces. As an example, we extend the theory of perfectoid spaces to this setting.
Products of $p$-Adic Valuation Trees
本研究围绕数论中的素因子整除性问题，利用$p$-进赋值构建可视化的$p$-进赋值树，探究有理系数线性函数及多项式乘积生成树的节点分裂规律，明确了这类多项式树的无限分支与终止节点赋值特征。
In number theory, we know Legendre's formula $ v_p(n!) = \sum_{k \ge 1} \lfloor \frac{n}{p^k} \rfloor $, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$. There is also the second (or alternative) equality $ v_p (n!) = \frac{n-s_p(n)}{p-1} $ where $s_p(n)$ is the $p$-adic weight of $n$ or the sum of digits of $n$ in base $p$. Both kinds of Legendre's formula allow us to determine valuations of the natural number, the odd factorial, binomial coefficients, Catalan numbers, and other combinatorial objects. The artic…
We study how the $\ell$-adic valuation of the number of spanning trees varies in regular abelian $\ell$-towers of multigraphs. We show that for an infinite family of regular abelian $\ell$-towers of bouquets, the behavior of the $\ell$-adic valuation of the number of spanning trees behave similarly to the $\ell$-adic valuation of the class numbers in $\mathbb{Z}_{\ell}$-extensions of number fields.
We compute the $p$-adic regulator of cyclic cubic extensions of $\mathbb Q$ with discriminant up to $10^{16}$ for $3<p<100$, and observe the distribution of the $p$-adic valuation of the regulators. We find that for almost all primes, the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model, a conjecture on the distribution of the valuations of $p$-adic regulators of cyclic cubic fields is stated.
For a prime $p$ and an integer $x$, the $p$-adic valuation of $x$ is denoted by $ν_{p}(x)$. For a polynomial $Q$ with integer coefficients, the sequence of valuations $ν_{p}(Q(n))$ is shown to be either periodic or unbounded. The first case corresponds to the situation where $Q$ has no roots in the ring of $p$-adic integers. In the periodic situation, the period length is determined.
The $p$-adic Valuations of Möbius Duals of Lucas Sequences
本文研究了Möbius双曲Lucas序列的p进估值，并探讨了其与Wall-Sun-Sun素数存在的关系。
In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both related to the special case $p = 2$. The crucial point for obtaining our main result is the fact that the $p$-adic valuation of the rational numbers in question is unbounded from above. We will confirm this fact by three different methods; the first two are elementary while the third one leans on the $p$-adic analysis.
Let $E_{-D}$ be the elliptic curve $y^2=x^3+Dx$ defined over $K=\mathbb{Q}(i)$ for $D\in K$ which is coprime to 2. In this paper, we give a lower bound for the 2-adic valuation of the algebraic part of the central value of Hecke $L$-function associated to $E_{-D}$.
We define the Narayana sequence $\{a_n\}_{n\geq 0}$ as the one satisfying the linear recurrence relation $a_n = a_{n-1}+a_{n-3}$ for $n\geq 3$, with initial values $a_0 = 0$ and $a_1 = a_2=1$. In this paper, we fully characterize the $3$-adic valuation of ${a_n}$ and use this to determine all Narayana numbers that are factorials.
We show that the $p$-adic valuation of the sequence of Fibonacci numbers is a $p$-regular sequence for every prime $p$. For $p \neq 2, 5$, we determine that the rank of this sequence is $α(p) + 1$, where $α(m)$ is the restricted period length of the Fibonacci sequence modulo $m$.
In this paper, we determine all the squares in the sequence $\{\prod_{k=2}^n(k^2-1)\}_{n=2}^\infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence.
For integers $k \geq 2$ and $n \neq 0$, let $v_k(n)$ denotes the greatest nonnegative integer $e$ such that $k^e$ divides $n$. Moreover, let $u_n$ be a nondegenerate Lucas sequence satisfying $u_0 = 0$, $u_1 = 1$, and $u_{n + 2} = a u_{n + 1} + b u_n$, for some integers $a$ and $b$. Shu and Yao showed that for any prime number $p$ the sequence $v_p(u_{n + 1})$ is $p$-regular, while Medina and Rowland found the rank of $v_p(F_{n + 1})$, where $F_n$ is the $n$-th Fibonacci number. We prove that if $k$ and $b$ are relatively prime then $v_k(u_{n + 1})$ is a $k$-regular sequence, and for $k$ a pri…
The $p$-Adic Valuation Trees for Quadratic Polynomials for Odd Primes
本文用树结构刻画奇素数下整系数二次多项式的 $p$-进赋值序列，并依据系数判定树的有限性、分支、层级及节点赋值。
For fixed integers $D \geq 0$ and $c \geq 3$, we demonstrate how to use $2$-adic valuation trees of sequences to analyze Diophantine equations of the form $x^2+D=2^cy$ and $x^3+D=2^cy$, for $y$ odd. Further, we show for what values $D \in \mathbb{Z}^+$, the numbers $x^3+D$ will generate infinite valuation trees, which lead to infinite solutions to the above Diophantine equations.
For all integers $n \geq k \geq 1$, define $H(n,k) := \sum 1 / (i_1 \cdots i_k)$, where the sum is extended over all positive integers $i_1 < \cdots < i_k \leq n$. These quantities are closely related to the Stirling numbers of the first kind by the identity $H(n,k) = s(n + 1, k + 1) / n!$. Motivated by the works of Erdős-Niven and Chen-Tang, we study the $p$-adic valuation of $H(n,k)$. In particular, for any prime number $p$, integer $k \geq 2$, and $x \geq (k-1)p$, we prove that $ν_p(H(n,k)) < -(k - 1)(\log_p(n/(k - 1)) - 1)$ for all positive integers $n \in [(k-1)p, x]$ whose base $p$ repre…
Following Sun and Moll, we study v_p(T(N)), the p-adic valuation of the counting function of the alternating sign matrices. We find an exact analytic expression for it that exhibits the fluctuating behaviour, by means of Fourier coefficients. The method is the Mellin-Perron technique, which is familiar in the analysis of the sum-of-digits function and related quantities.
In this paper we compute the 2-adic valuations of some polynomials associated with the definite integral $\int_{0}^{\infty} \frac{dx}{(x^4+2*a*x^2+1)^(m+1)}$
Properties of 2-adic valuation sequences for general quadratic polynomials with integer coefficients are determined directly from the coefficients. These properties include boundedness or unboundedness, periodicity, and valuations at terminating nodes. We completely describe the periodic sequences in the bounded case. Throughout, we frame results in terms of trees and sequences.
In this note, basing on a certain functional equation of the dilogarithm function, we establish nontrivial lower bounds for the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers involving harmonic numbers. Then we use our estimate to derive the integrality of some sequences of rational numbers, which cannot be seen directly from their definitions.
On the $p$-adic valuations of values of Legendre polynomials
本文研究 Legendre 多项式在素数处取值的 $p$-进赋值，给出显式公式，推广相关旧猜想并解决 Cigler 于 2017 年提出的问题。
In this paper we focus on finding all the factorials expressible as a product of a fixed number of $2k$-nacci numbers with $k \geq 2$. We derive the 2-adic valuation of the $2k$-nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with $p$-regular sequences.
In this paper will be proved an inequality regarding $v_2(a^{b}-c^{d})$. Using this formula it will be possible to have informations about the divisibility of 2 of this function without computing it. Then, will be studied the behavior of this function when it approaches infinity and there will be a lot of analogies with 2-adic integers. This result may be easily used on a calculator, to facilitate the analysis of the divisibility by 2 of this function for large values. Furthermore 2-adic valuation and in general 2-adic analysis is related to binary system, and every work related to it could ha…
Stability of $p$-adic valuations of Hecke L-values
本文证明了 CM 域上代数 Hecke 特征在 $\ell$-进反循环扭曲族（$\ell\neq p$）中临界 L-值的 $p$-稳定性结果。
Let $k\in\N_{\geq 2}$ and for given $m\in\Z\setminus\{0\}$ consider the sequence $(S_{k,m}(n))_{n\in\N}$ defined by the power series expansion $$ \frac{1}{(1-x)^{m}}\prod_{i=0}^{\infty}\frac{1}{(1-x^{k^{i}})^{m}}=\sum_{n=0}^{\infty}S_{k,m}(n)x^{n}. $$ The number $S_{k,m}(n)$ for $m\in\N_{+}$ has a natural combinatorial interpretation: it counts the number of representations of $n$ as sums of powers of $k$, where the part equal to $1$ takes one among $mk$ colors and each part $>1$ takes $m(k-1)$ colors. We concentrate on the case when $k=p\in\mathbb{P}$. Our main result is the computation of th…
The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes
本文证明了虚二次域在惯性素数处沿反分圆塔的Hecke $L$-值 $p$-adic 赋值的渐近公式，通过确定广义高斯和与局部预解式的 $p$-adic 赋值，回答了 Rubin 的问题。
$p$-adic valuation of harmonic sums and their connections with Wolstenholme primes
本研究围绕Eswarathasan-Levine关于调和数$p$进赋值的猜想展开，用模算术方法分别证明了非Wolstenholme素数与Wolstenholme素数情形下，$H(p^mn)$的$p$进赋值在$H(n)$的$p$进赋值为3时的反常渐近行为。
A note on the $q$-adic valuation of $σ_k(n)$
本文通过LTE引理和 cyclotomic多项式，推导出σ_k(n)的q进位的精确公式，并给出优于 Zhao 之前结果的渐进行为上界。
In this article, we give explicit formulas for the $p$-adic valuations of the Fibonomial coefficients ${p^a n \choose n}_F$ for all primes $p$ and positive integers $a$ and $n$. This is a continuation from our previous article extending some results in the literature, which deal only with $p = 2,3,5,7$ and $a = 1$. Then we use these formulas to characterize the positive integers $n$ such that ${pn \choose n}_F$ is divisible by $p$, where $p$ is any prime which is congruent to $\pm 2 \pmod{5}$.
The 2-adic valuations of the algebraic central $L$-values for quadratic twists of weight 2 newforms
本文在温和假设下，给出了权2新形式的二次扭曲的代数中心$L$-值的2-adic赋值的尖锐下界，并对无穷多个导子计算了该赋值。
Fix distinct primes $\ell$ and $f$, a finite field $\mathbf{F}_{q}$ such that $q \equiv 1 \pmod{\ell f}$, and a character $χ: \mathbf{F}_{q}^{\times} \to \mathbf{C}^{\times}$ of exact order $\ell f$. We present a new $\ell$-adic congruence for the Jacobi sum $J(χ^{\ell}, χ^{f})$. These Jacobi sums are Frobenius eigenvalues of the curve $y^{\ell} = x^{f} + 1$.
$L-$series attached to two classical families of elliptic curves with complex multiplications are studied over number fields, formulae for their special values at $s=1, $ bound of the values, and criterion of reaching the bound are given. Let $ E_1: y^{2}=x^{3}-D_1 x $ be elliptic curves over the Gaussian field $K=\Q(\sqrt{-1}), $ with $ D_1 =π_{1} ... π_{n} $ or $ D_1 =π_{1} ^{2}... π_{r} ^{2} π_{r+1} ... π_{n}$, where $π_{1}, ..., π_{n}$ are distinct primes in $K$. A formula for special values of Hecke $L-$series attached to such curves expressed by Weierstrass $\wp-$function are given; a lo…
In this document will be proved a formula to compute the $p$-adic valuation of a hyperfactorial. We call a hyperfactorial the result of multiplying a given number of consecutive integers from 1 to the given number,each raised to its own power. For example, the hyperfactorial of $n$ is equal to: $1^1 2^2 3^3\dots n^n$ . Lots of studies have been done about the hyperfactorial function, in particular two mathematicians: Glaisher and Kinkelin, who have found the asymptotic behaviour of this function as $n$ that approaches infinity (finding a costant, the Glaisher-Kinkelin costant, which has a lot …
Let $m$ and $n>0$ be integers. Suppose that $p$ is a prime dividing $m-4$ but not dividing $m$. We show that $ν_p(\sum_{k=0}^{n-1}\frac{\binom{2k}k}{m^k})$ and $ν_p(\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k})$ are at least $ν_p(n)$, where $ν_p(x)$ denotes the $p$-adic valuation of $x$. Furthermore, if $p>3$ then $$n^{-1}\sum_{k=0}^{n-1}\frac{\bi{2k}k}{m^k}=\frac{\binom{2n-1}{n-1}}{4^{n-1}} (mod p^{ν_p(m-4)})$$ and $$n^{-1}\sum_{k=0}^{n-1}\binom{n-1}{k}(-1)^k\frac{\binom{2k}k}{m^k}=\frac{C_{n-1}}{4^{n-1}} (mod p^{ν_p(m-4)}),$$ where $C_k$ denotes the Catalan number $\binom{2k}{…
Bounds on the $p$-adic valuation of the factorial, hyperfactorial and superfactorial
本文利用 Legendre-de Polignac 公式研究了阶乘、超阶乘和超级阶乘等量的 p 进赋值的上下界，并讨论了其在密码学中的潜在应用以及 Stirling 数和 Catalan 数的相关界。
Let $m, n, k$ and $c$ be positive integers. Let $ν_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of the second kind and provide a detailed 2-adic analysis to the Stirling numbers of the second kind. Consequently, we show that if $2\le m\le n$ and $c$ is odd, then $ν_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except when $n=m=2$ and $c=1$, in which case $ν_2(S(8,3)-S(4,3))=6$. This solves a conjecture of Lengyel proposed in 2009.
On non-holonomicity, transcendence and $p$-adic valuations
本文研究了 $p$-进赋值相关幂级数的性质，证明其在特征 0 下非完整且非代数，找到了使级数值超越的无穷多个有理数与代数无理数，并应用于 $p$-自动序列。
In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.
The $p$-adic Galois Cohomology of Valuation Fields
本文研究预完美oid域的 $p$-进赋值域扩张，利用 Gabber-Ramero 的余切复形计算其 Galois 上同调，并推广且重新证明 Tate–Hyodo 经典定理。
In this paper, we will study p-adic q-expansion of alternating sums of powers. From these properties, we derive some interesting properties related to p-adic q-expansion of alternating sums of powers
Let $b \geq 2$ be an integer base with prime factors $p_1, \ldots, p_s$. In this paper we study sequences of "$b$-adic valuations" and last nonzero digits in $b$-adic expansions of the values $f(n) = (f_1(n), \ldots, f_s(n))$, where each $f_i$ is a $p_i$-adic analytic function. We give a complete classification concerning $k$-regularity of these sequences, which generalizes a result for $b$ prime obtained by Shu and Yao. As an application, we strengthen a theorem by Murru and Sanna on $b$-adic valuations of Lucas sequences of the first kind. Moreover, we derive a method to determine precisely …
We generalize results on the $p$-adic valuations of $S(n,k)$, the Stirling number of the second kind and $s(n,k)$ the Stirling number of the first kind. We have several new estimates for these valuations, along with criteria for when the estimates are sharp. The primary foci are the explicit evaluation of $ν_2(S(n,k))$ with $n=c2^h$, $k=b2^h+a$, $a, b, c, h, k, n \in Z^+$, and $1\le a \le 2^{h-1}$, and $ν_p(S(n,k))$ when $n=cp^h$ for an odd prime $p$. We have strong new results, which generalize and strengthen previous results, for all primes. We also have some new results on the $p$-adic valu…
The Shub-Smale Tau Conjecture is a hitherto unproven statement (on integer roots of polynomials) whose truth implies both a variant of $P\neq NP$ (for the BSS model over C) and the hardness of the permanent. We give alternative conjectures, some potentially easier to prove, whose truth still implies the hardness of the permanent. Along the way, we discuss new upper bounds on the number of $p$-adic valuations of roots of certain sparse polynomial systems, culminating in a connection between quantitative p-adic geometry and complexity theory.
Let $\mathscr{O}_K$ be a 2-adic discrete valuation ring with perfect residue field $k$. We classify $p$-divisible groups and $p$-power order finite flat group schemes over $\mathscr{O}_K$ in terms of certain Frobenius module over $\mathfrak{S}:=W(k)[[u]]$. We also show the compatibility with crystalline Dieudonné theory and associated Galois representations. Our approach differs from Lau's generalization of display theory, and we additionally obtain the the compatibility with associated Galois representations.
Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ the 2-adic valuation of $n$. The Stirling numbers of the first kind, denoted by $s(n,k)$, counts the number of permutations of $n$ elements with $k$ disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the $p$-adic valuations of $s(n,k)$. In this paper, by introducing the concept of $m$-th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of $s(2^n, k)$. We also prove that $v_2(s(2^n+1,k+1))=v_…
On the $p$-adic valuation of third order linear recurrence sequences
本文通过推广Marques和Lengyel的猜想，研究了三阶线性递推序列的$p$-进赋值，证明了若猜想对特定序列成立，则丢番图方程$x_n=m!$的解集有限且可被有效计算。
Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Assume that $v_p(f(n))\ge s_1$ and $v_p(g(n))\ge s_2$ hold for all integers $n$ for some $s_1, s_2$ fixed non-negative integers. Let $S$ denote the maximum of $v_p(gcd(f(n),g(n)))$ over all integers $n$. In this paper, we establish multiple lower bound for $v_p(r)$. More specifically, we show that $v_p(r)\ge S-\max{s_1,s_2}+ps_1s_2\frac{p-1}{p-p^{-k}}$, where $k=\lfloor \log_p((p-1)\max\{s_1,s_2\}+1)\rfloor -1$.
We investigate the $p$-adic valuation of Weil sums of the form $W_{F,d}(a)=\sum_{x \in F} ψ(x^d -a x)$, where $F$ is a finite field of characteristic $p$, $ψ$ is the canonical additive character of $F$, the exponent $d$ is relatively prime to $|F^\times|$, and $a$ is an element of $F$. Such sums often arise in arithmetical calculations and also have applications in information theory. For each $F$ and $d$ one would like to know $V_{F,d}$, the minimum $p$-adic valuation of $W_{F,d}(a)$ as $a$ runs through the elements of $F$. We exclude exponents $d$ that are congruent to a power of $p$ modulo …
Transcendence and algebraic independence of a family of $p$-adic valuation generating functions
本文研究了p进估值生成函数族的超越性与代数独立性，证明了其在单位圆内非零代数点处的超越性，并得出了乘法独立代数参数下的代数独立性结果。
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