{"title":"On Relations Associated with the Euler Function","authors":"V. K. Leontiev, E. N. Gordeev","doi":"10.1134/S1990478923040075","DOIUrl":null,"url":null,"abstract":"<p> The paper studies the properties of the set of numbers smaller than and coprime to\n<span>\\( n \\)</span> with the modulo\n<span>\\( n \\)</span> multiplication operation introduced on it (this object is sometimes called the\nEuler group). The cardinality of such a set is the well-known Euler function\n<span>\\( \\varphi (n) \\)</span>, which is one of the classical functions in the number theory. The fields of its\napplication are quite wide and include, for example, various branches of discrete mathematics, and\nit also has significant applications in cryptography. The paper considers various combinatorial\nproblems arising in the study of the Euler group and the Euler function. Relations between\ntheoretical and numerical parameters associated with the Euler group and Euler function are\nderived. The combinatorial relations obtained in the paper can be used when solving applied\ncombinatorial problems and in cryptography.\n</p>","PeriodicalId":607,"journal":{"name":"Journal of Applied and Industrial Mathematics","volume":"17 4","pages":"760 - 766"},"PeriodicalIF":0.5800,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied and Industrial Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1134/S1990478923040075","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Engineering","Score":null,"Total":0}
引用次数: 0
Abstract
The paper studies the properties of the set of numbers smaller than and coprime to
\( n \) with the modulo
\( n \) multiplication operation introduced on it (this object is sometimes called the
Euler group). The cardinality of such a set is the well-known Euler function
\( \varphi (n) \), which is one of the classical functions in the number theory. The fields of its
application are quite wide and include, for example, various branches of discrete mathematics, and
it also has significant applications in cryptography. The paper considers various combinatorial
problems arising in the study of the Euler group and the Euler function. Relations between
theoretical and numerical parameters associated with the Euler group and Euler function are
derived. The combinatorial relations obtained in the paper can be used when solving applied
combinatorial problems and in cryptography.
期刊介绍:
Journal of Applied and Industrial Mathematics is a journal that publishes original and review articles containing theoretical results and those of interest for applications in various branches of industry. The journal topics include the qualitative theory of differential equations in application to mechanics, physics, chemistry, biology, technical and natural processes; mathematical modeling in mechanics, physics, engineering, chemistry, biology, ecology, medicine, etc.; control theory; discrete optimization; discrete structures and extremum problems; combinatorics; control and reliability of discrete circuits; mathematical programming; mathematical models and methods for making optimal decisions; models of theory of scheduling, location and replacement of equipment; modeling the control processes; development and analysis of algorithms; synthesis and complexity of control systems; automata theory; graph theory; game theory and its applications; coding theory; scheduling theory; and theory of circuits.