Arithmetic properties for Fu's 9 dots bracelet partitions

2015 ◽  
Vol 11 (04) ◽  
pp. 1063-1072
Author(s):  
Olivia X. M. Yao
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Arithmetic Properties ◽  
Powers Of 2

The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let 𝔅k(n) denote the number of Fu's k dots bracelet partitions of n. Fu also proved several congruences modulo primes and modulo powers of 2. Recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for 𝔅5(n), 𝔅7(n) and 𝔅11(n). More recently, Cui and Gu, and Xia and the author derived a number of congruences modulo powers of 2 for 𝔅5(n). In this paper, we prove four congruences modulo 2 and two congruences modulo 4 for 𝔅9(n) by establishing the generating functions of 𝔅9(An+B) modulo 4 for some values of A and B.

scholarly journals Arithmetic Properties of Overpartition Pairs into Odd Parts

10.37236/2274 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Lishuang Lin
Keyword(s):  
Positive Integer ◽  
Arithmetic Properties ◽  
Powers Of 2 ◽  
Fixed Positive Integer ◽  
Almost All ◽  
Small Powers

In this work, we investigate various arithmetic properties of the function $\overline{pp}_o(n)$, the number of overpartition pairs of $n$ into odd parts. We obtain a number of Ramanujan type congruences modulo small powers of $2$ for $\overline{pp}_o(n)$. For a fixed positive integer $k$, we further show that $\overline{pp}_o(n)$ is divisible by $2^k$ for almost all $n$. We also find several infinite families of congruences for $\overline{pp}_o(n)$ modulo $3$ and two formulae for $\overline{pp}_o(6n+3)$  and  $\overline{pp}_o(12n)$ modulo $3$.

scholarly journals ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

2008 ◽  
Vol 78 (1) ◽  
pp. 129-140 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Quadratic Forms ◽  
Triangular Numbers ◽  
Representations Of Integers

AbstractGenerating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

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.

A NEW CONGRUENCE MODULO 25 FOR 1-SHELL TOTALLY SYMMETRIC PLANE PARTITIONS

2014 ◽  
Vol 91 (1) ◽  
pp. 41-46 ◽  
Author(s):  
ERNEST X. W. XIA
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Plane Partitions ◽  
Arithmetic Properties ◽  
Symmetric Plane ◽  
Congruence Modulo

AbstractFor any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$

An inequality for characteristic functions

1972 ◽  
Vol 6 (1) ◽  
pp. 1-9 ◽  
Author(s):  
C.R. Heathcote ◽  
J.W. Pitman
Keyword(s):  
Positive Integer ◽  
Characteristic Function ◽  
Characteristic Functions ◽  
The Real ◽  
Powers Of 2

The paper is concerned with an extension of the inequality 1 - u(2nt) ≤ 4n[1-u(t)] for u(t) the real part of a characteristic function. The main result is that the inequality in fact holds for all positive integer n and not only powers of 2. Certain consequences are deduced and a brief discussion is given of the circumstances under which equality holds.

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.

Arithmetic properties of a partition pair function

2014 ◽  
Vol 10 (06) ◽  
pp. 1583-1594 ◽  
Author(s):  
Shi-Chao Chen
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Arithmetic Properties ◽  
Pair Function

For a positive integer n, let ped (n) be the number of partitions of n where the even parts are distinct, and [Formula: see text] be the number of overpartitions of n into odd parts. Moreover, let Q(n) denote the number of the partition pairs of n into two colors (say, red and blue), where the parts colored red satisfy restrictions of partitions counted by ped (n), while the parts colored blue satisfy restrictions of partitions counted by [Formula: see text]. We establish several congruences for Q(n). We also obtain an asymptotic formula for Q(n).

scholarly journals Arithmetic properties for Andrews’ (48,6)- and (48,18)-singular overpartitions

2019 ◽  
Vol 17 (1) ◽  
pp. 356-364
Author(s):  
Eric H. Liu ◽  
Wenjing Du
Keyword(s):  
Arithmetic Properties ◽  
Powers Of 2

Abstract Singular overpartition functions were defined by Andrews. Let Ck,i(n) denote the number of (k, i)-singular overpartitions of n, which counts the number of overpartitions of n in which no part is divisible by k and only parts ±i (mod k) may be overlined. A number of congruences modulo 3, 9 and congruences modulo powers of 2 for Ck,i(n) were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pairs (k, i). In this paper, we prove some congruences modulo powers of 2 for C48, 6(n) and C48, 18(n).

scholarly journals Arithmetic properties of polynomial solutions of the Diophantine equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$

2021 ◽  
Author(s):  
Karl Dilcher ◽  
Maciej Ulas
Keyword(s):  
Differential Equations ◽  
Diophantine Equation ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Polynomial Solutions ◽  
Arithmetic Properties

AbstractFor each integer $$n\ge 1$$ n ≥ 1 we consider the unique polynomials $$P, Q\in {\mathbb {Q}}[x]$$ P , Q ∈ Q [ x ] of smallest degree n that are solutions of the equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$ P ( x ) x n + 1 + Q ( x ) ( x + 1 ) n + 1 = 1 . We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.

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.

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