New Concepts of Congruences Modulo 1/n and Positive Integers Modulo 1/n
Habib Lebsir
Habib Lebsir, Cité Salah Boulkeroua Bt8n8-Skikda, Algeria.
Manuscript received on 16 January 2025 | First Revised Manuscript received on 26 January 2025 | Second Revised Manuscript received on 16 March 2025 | Manuscript Accepted on 15 April 2025 | Manuscript published on 30 April 2025 | PP: 11-21 | Volume-5 Issue-1, April 2025 | Retrieval Number: 100.1/ijam.A118805010425 | DOI: 10.54105/ijam.A1188.05010425
Open Access | Editorial and Publishing Policies | Cite | Zenodo | OJS | Indexing and Abstracting
© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: The aim of this article is the creation of new Mathematical beings: The positive integers Modulo 1/n. I hope that they will be used by all the scientific community in general and the computer scientists (cryptographers) particularly. In the other hand, the paper contributes to enlarge the field of number theory and algebraic sets. In this paper, I present a new type of congruences on IN: the congruences modulo 1/N and The Sets of positive integers modulo 1/N denoted respectively = [1/N] AND IN 1/n, n € IN* The Work is divided into two Sections : Section A: It’s composed of: ▪ A fundamental Theorem (With Proof) ▪ A fundamental relation of equivalence on IN denoted = [1/n]
▪ The definition of the sets IN1/n , n € IN* (with examples)
▪ The definition of the binary operations*and ∆ on IN
▪ The definition of the binary operations *and ∆ on IN 1/n, n € IN*
▪ The presentation of the semi-groups (IN ; *) ; (IN ; ∆) ; ( IN1/n ;*); ( IN 1/n ; ∆ Section B: It’s Composed of an Appendice with: I) The definition and graph of the function: r: IN* → IN* x → n0 Where n0 is the rank of x . II) The definition and graph of the function: f: IN* → IN* n |→ [100, 1/n]
Keywords: Congruences Modulo, Positive integers Modulo, Integer Part, Partition, Equivalence Classes, Rank, Amplitude, Semi-group.
Scope of the Article: Algebra