The2-adic and3-adic valuation of the Tripell sequence and an application

2020 ◽  
Vol 44 (1) ◽  
pp. 131-141
Author(s):  
Jhon Jairo BRAVO ◽  
Maribel DÍAZ ◽  
José Luis RAMÍREZ
Keyword(s):  
Adic Valuation

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

scholarly journals 2-Adic valuations of Stirling numbers of the first kind

2019 ◽  
Vol 15 (09) ◽  
pp. 1827-1855 ◽  
Author(s):  
Min Qiu ◽  
Shaofang Hong
Keyword(s):  
Explicit Formula ◽  
Symmetric Functions ◽  
Stirling Numbers ◽  
Elementary Symmetric Functions ◽  
Disjoint Cycles ◽  
Adic Valuation ◽  
Positive Integers

Let [Formula: see text] and [Formula: see text] be positive integers. We denote by [Formula: see text] the 2-adic valuation of [Formula: see text]. The Stirling numbers of the first kind, denoted by [Formula: see text], count the number of permutations of [Formula: see text] elements with [Formula: see text] disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the study of the [Formula: see text]-adic valuations of [Formula: see text]. In this paper, by introducing the concept of [Formula: see text]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 [Formula: see text]. We also prove that [Formula: see text] holds for all integers [Formula: see text] between 1 and [Formula: see text]. As a corollary, we show that [Formula: see text] if [Formula: see text] is odd and [Formula: see text]. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if [Formula: see text], then [Formula: see text] and [Formula: see text], where [Formula: see text] stands for the [Formula: see text]th elementary symmetric functions of [Formula: see text]. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.

scholarly journals Further Results on a Curious Arithmetic Function

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Long Chen ◽  
Kaimin Cheng ◽  
Tingting Wang
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Function ◽  
Rational Numbers ◽  
Arithmetic Properties ◽  
Adic Valuation ◽  
Positive Integers

Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

DISTRIBUTION OF BINOMIAL COEFFICIENTS AND DIGITAL FUNCTIONS

2001 ◽  
Vol 64 (3) ◽  
pp. 523-547 ◽  
Author(s):  
GUY BARAT ◽  
PETER J. GRABNER
Keyword(s):  
Arithmetic Functions ◽  
Binomial Coefficients ◽  
Asymptotic Results ◽  
Residue Classes ◽  
Adic Valuation

The distribution of binomial coefficients in residue classes modulo prime powers and with respect to the p-adic valuation is studied. For this purpose, general asymptotic results for arithmetic functions depending on blocks of digits with respect to q-ary expansions are established.

scholarly journals The 2-adic Valuation of Stirling Numbers

2008 ◽  
Vol 17 (1) ◽  
pp. 69-82 ◽  
Author(s):  
Tewodros Amdeberhan ◽  
Dante Manna ◽  
Victor H. Moll
Keyword(s):  
Stirling Numbers ◽  
Adic Valuation

scholarly journals SQUARES IN (22 - 1)…(n2 - 1) AND p-ADIC VALUATION

2010 ◽  
Vol 03 (02) ◽  
pp. 329-333 ◽  
Author(s):  
Shaofang Hong ◽  
Xingjiang Liu
Keyword(s):  
Adic Valuation

In this paper, we determine all the squares in the sequence [Formula: see text]. 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.

scholarly journals The 2-adic valuation of the cardinality of Jacobians of genus 2 curves over quadratic towers of finite fields

2019 ◽  
Vol 18 (07) ◽  
pp. 1950135
Author(s):  
Ricard Garra ◽  
Josep M. Miret ◽  
Jordi Pujolàs ◽  
Nicolas Thériault
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Sylow Subgroup ◽  
Quadratic Extensions ◽  
Genus 2 ◽  
Adic Valuation ◽  
Genus 2 Curves

Given a genus 2 curve [Formula: see text] defined over a finite field [Formula: see text] of odd characteristic such that [Formula: see text], we study the growth of the 2-adic valuation of the cardinality of the Jacobian over a tower of quadratic extensions of [Formula: see text]. In the cases of simpler regularity, we determine the exponents of the 2-Sylow subgroup of [Formula: see text].

scholarly journals On the p-adic valuation of stirling numbers of the first kind

2016 ◽  
Vol 151 (1) ◽  
pp. 217-231 ◽  
Author(s):  
P. Leonetti ◽  
C. Sanna
Keyword(s):  
Stirling Numbers ◽  
Adic Valuation
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