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.

NOTES ON A CURIOUS ARITHMETIC FUNCTION

2011 ◽  
Vol 04 (04) ◽  
pp. 705-714 ◽  
Author(s):  
Yuanming Zhong ◽  
Qianrong Tan
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Arithmetic Function ◽  
Product Formula ◽  
Adic Valuation ◽  
Positive Integers

Let v2(n) denote the 2-adic valuation of any positive integer n. Recently, Farhi introduced a curious arithmetic function f defined for any positive integer n by [Formula: see text]. Farhi showed that the inequality [Formula: see text] with c = 4.01055487… holds for all positive integer n and conjectured that one can replace the upper bound cn by 4n in this inequality. In this paper, we show two identities about the product [Formula: see text] and then use it to prove partially Farhi's conjecture. Finally, we propose a conjecture from which the truth of Farhi's conjecture can be deduced. In particular, we confirm the truth of our conjecture for all positive integers n up to 100000 by using Matlab 7.1.

NOTE ON LEHMER–PIERCE SEQUENCES WITH THE SAME PRIME DIVISORS

2017 ◽  
Vol 97 (1) ◽  
pp. 11-14
Author(s):  
M. SKAŁBA
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Prime Divisors ◽  
Positive Integers

Let $a_{1},a_{2},\ldots ,a_{m}$ and $b_{1},b_{2},\ldots ,b_{l}$ be two sequences of pairwise distinct positive integers greater than $1$. Assume also that none of the above numbers is a perfect power. If for each positive integer $n$ and prime number $p$ the number $\prod _{i=1}^{m}(1-a_{i}^{n})$ is divisible by $p$ if and only if the number $\prod _{j=1}^{l}(1-b_{j}^{n})$ is divisible by $p$, then $m=l$ and $\{a_{1},a_{2},\ldots ,a_{m}\}=\{b_{1},b_{2},\ldots ,b_{l}\}$.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

SUMS OF SQUARES AND PARTITION CONGRUENCES

2020 ◽  
pp. 1-19
Author(s):  
SU-PING CUI ◽  
NANCY S. S. GU
Keyword(s):  
Positive Integer ◽  
Sums Of Squares ◽  
Binomial Coefficients ◽  
Partition Congruences ◽  
Arithmetic Properties ◽  
Positive Integers ◽  
Pair Function

For positive integers $n$ and $k$ , let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$ , by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$ . Furthermore, in view of these arithmetic properties of $r_{k}(n)$ , we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

scholarly journals NUMBER THEORETIC PROPERTIES OF WRONSKIANS OF ANDREWS–GORDON SERIES

2008 ◽  
Vol 04 (02) ◽  
pp. 323-337 ◽  
Author(s):  
ANTUN MILAS ◽  
ERIC MORTENSON ◽  
KEN ONO
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Positive Integer ◽  
Minimal Models ◽  
Characteristic P ◽  
Partition Identities ◽  
Arithmetic Properties ◽  
Conformal Field ◽  
Supersingular Elliptic Curves ◽  
Positive Integers

For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews–Gordon q-series [Formula: see text] This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of [Formula: see text] Virasoro minimal models. We determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if Pb(a;n) denotes the number of partitions of n into parts which are not congruent to 0, ±a ( mod b), then for every positive integer n, we have [Formula: see text] We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k + 1 = p, where p ≥ 5 is prime, and the quotient is non-zero, then it is essentially the locus of supersingular j-invariants in characteristic p.

scholarly journals Nontrivial upper bounds for the least common multiple of an arithmetic progression

2020 ◽  
pp. 2150138
Author(s):  
Sid Ali Bousla
Keyword(s):  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Function ◽  
Upper Bounds ◽  
Addition Formula ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Finite Arithmetic ◽  
Positive Integers ◽  
Coprime Positive Integers

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], we have [Formula: see text] where [Formula: see text]. If in addition [Formula: see text] is a prime number and [Formula: see text], then we prove that for any [Formula: see text], we have [Formula: see text], where [Formula: see text]. Finally, we apply those inequalities to estimate the arithmetic function [Formula: see text] defined by [Formula: see text] ([Formula: see text]), as well as some values of the generalized Chebyshev function [Formula: see text].

scholarly journals On simultaneous rational approximation to a p-adic number and its integral powers

2011 ◽  
Vol 54 (3) ◽  
pp. 599-612 ◽  
Author(s):  
Yann Bugeaud ◽  
Natalia Budarina ◽  
Detta Dickinson ◽  
Hugh O'Donnell
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Rational Approximation ◽  
Real Numbers ◽  
The Real ◽  
Positive Integers ◽  
Simultaneous Rational Approximation

AbstractLet p be a prime number. For a positive integer n and a p-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖p are all less than q−λ−1. Here, ‖x‖p denotes the infimum of |x−n|p as n runs through the integers. We study the set of values taken by the function λn.

scholarly journals 3x+1 Minus the +

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Kenneth G. Monks
Keyword(s):  
Positive Integer ◽  
Rational Numbers ◽  
General Mechanism ◽  
Arbitrary Positive Integer ◽  
Arbitrary Integer ◽  
Constant Multiple ◽  
International Audience ◽  
Positive Integers

International audience We use Conway's \emphFractran language to derive a function R:\textbfZ^+ → \textbfZ^+ of the form R(n) = r_in if n ≡ i \bmod d where d is a positive integer, 0 ≤ i < d and r_0,r_1, ... r_d-1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the R-orbit of 2^n contains 2 for all positive integers n. We then show that the R-orbit of an arbitrary positive integer is a constant multiple of an orbit that contains a power of 2. Finally we apply our main result to show that any cycle \ x_0, ... ,x_m-1 \ of positive integers for the 3x+1 function must satisfy \par ∑ _i∈ \textbfE \lfloor x_i/2 \rfloor = ∑ _i∈ \textbfO \lfloor x_i/2 \rfloor +k. \par where \textbfO=\ i : x_i is odd \ , \textbfE=\ i : x_i is even \ , and k=|\textbfO|. \par The method used illustrates a general mechanism for deriving mathematical results about the iterative dynamics of arbitrary integer functions from \emphFractran algorithms.

scholarly journals The least common multiple of consecutive arithmetic progression terms

2011 ◽  
Vol 54 (2) ◽  
pp. 431-441 ◽  
Author(s):  
Shaofang Hong ◽  
Guoyou Qian
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Arithmetic Progression ◽  
Arithmetic Function ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Image Position ◽  
Positive Integers

AbstractLet k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n byIf we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

scholarly journals On Two Problems Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3273
Author(s):  
Pavel Trojovský
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Function ◽  
Fibonacci Sequence ◽  
Infinitely Many Solutions ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Divisibility Properties

The order of appearance (in the Fibonacci sequence) function z:Z≥1→Z≥1 is an arithmetic function defined for a positive integer n as z(n)=min{k≥1:Fk≡0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn→∞(z(n+1)−z(n))/(logn)2−ϵ=∞, for all ϵ∈(0,2).

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