Arithmetic properties of (2, 3)-regular overcubic bipartitions

PurposeLet b¯2,3(n), which enumerates the number of (2, 3)-regular overcubic bipartition of n. The purpose of the paper is to describe some congruences modulo 8 for b¯2,3(n). For example, for each α ≥ 0 and n ≥ 1, b¯2,3(8n+5)≡0(mod8), b¯2,3(2⋅3α+3n+4⋅3α+2)≡0(mod8).Design/methodology/approachH.C. Chan has studied the congruence properties of cubic partition function a(n), which is defined by ∑n=0∞a(n)qn=1(q;q)∞(q2;q2)∞.FindingsTo establish several congruence modulo 8 for b¯2,3(n), here the author keeps to the classical spirit of q-series techniques in the proofs.Originality/valueThe results established in the work are extension to those proved in ℓ-regular cubic partition pairs.

RP-179: Formulation of Solutions of Standard Bi-Quadratic Congruence Modulo a POSITIVE INTEGER MULTIPLE to nth Power of Four

In this paper, the author has formulated the solutions of the standard bi-quadratic congruence of an even composite modulus modulo a positive integer multiple to nth power of four. First time a formula is established for the solutions. No literature is available for the current congruence. The author analysed the formulation of solutions in two different cases. In the first case of analysis, the congruence has the formulation which gives exactly eight incongruence solutions while in the second case of the analysis, the congruence has a different formulation of solutions and gives thirty-two incongruent solutions. A very simple and easy formulation to find all the solutions is presented here. Formulation is the merit of the paper.

How to predict good days in farming: ethnomathematics study with an ethnomodelling approach

Mathematics cannot be separated from everyday life. The use of mathematical concepts in cultural activities can be studied through the ethnomathematics program. However, ethnomathematics research may not be able to provide noticeable results, especially in constructing mathematical modelling for pedagogical purposes. Ethnomodelling later became one of the concepts introduced as an approach in ethnomathematics research. Based on the cultural aspect, the ability to predict a good day in farming is included in the holistic concept of culture because it belongs to the knowledge system and belief system (religion) in the universal element of culture. The research was conducted using an ethnomethodological approach and a realist ethnographic design. Based on this, this research was conducted to describe the ability of the Cigugur indigenous people in Kuningan Regency to predict what days are considered good to start farming activities. Data were collected by using observation techniques, in-depth interviews, documentation, and field notes. Data analysis techniques are carried out in stages through content analysis, triangulation, and pattern search. Based on the study of ethnomathematics, research that is able to describe the mathematical ideas and practices of the indigenous Cigugur community can be classified into several fundamental mathematical dimensions including counting, finding, measuring, designing, and explaining. The use of the ethnomodelling approach in research can describe several mathematical concepts used by the concepts of numbers, sets, relations, congruence, modulo, and mathematical modelling.

On the number of unit solutions of cubic congruence modulo $ n $

<abstract><p>For any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence $ x_1+\cdots+x_m\equiv c\pmod n $ with $ x_1, \ldots, x_m\in\mathbb{Z}_n^{\times} $ and gave a formula of its number of incongruent solutions. Recently, Taki Eldin extended A. Brauer's result to the quadratic case. In this paper, for any positive integer $ n $, we give an explicit formula for the number of incongruent solutions of the following cubic congruence</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $\end{document} </tex-math></disp-formula></p> </abstract>

A further q-analogue of Van Hamme’s (H.2) supercongruence for primes p ≡ 3(mod4)

Long and Ramakrishna [Some supercongruences occurring in truncated hyper- geometric series, Adv. Math. 290 (2016) 773–808] generalized the (H.2) supercongruence of Van Hamme to the modulus [Formula: see text] case. In this paper, we give a [Formula: see text]-analogue of Long and Ramakrishna’s result for [Formula: see text]. A [Formula: see text]-congruence modulo the fourth power of a cyclotomic polynomial, which is a deeper [Formula: see text]-analogue of the (A.2) supercongruence of Van Hamme for [Formula: see text], is also formulated.

Encryption and Decryption of a Message Involving Byte Rotation Technique and Invertible Matrix

The aim of this paper is to introduce a new encryption algorithm involving byte rotation and invertible matrix. In the proposed algorithm firstly we apply byte rotation to get an intermediate cipher and then applying the invertible matrix (modulo 27), which gives the final cipher text. Using secret key matrix along with congruence modulo, the message can be encrypted and decrypted perfectly.

Kamal Transformation based Cryptographic Technique in Network Security Involving ASCII Value

From old times, information security has been an essential part of human life. Day to day developing new technologies and increasing number of users share information with each other, securing information is becoming more important. For securing information, there are several techniques out of which most common is cryptography. Cryptography is a technique involving securing transmission of messages in presence of adversaries. The basic aim is to hide the information from unauthorized users. Cryptography includes to phases: encryption of plain message and decrypting encrypted message. The aim of this paper is to use new integral transform "Kamal transform" and congruence modulo operator involving ASCII value for encryption and decryption of message.