Solving some specific tasks by Euler's and Fermat's Little theorem

Authors

DOI:

https://doi.org/10.23755/rm.v37i0.485

Keywords:

Euler's theorem, coding, Fermat's Little theorem, linear congruences, cryptology, primality testing, Matlab

Abstract

Euler's and Fermat's Little theorems have a great use in number theory. Euler's theorem is currently widely used in computer science and cryptography, as one of the current encryption methods is an exponential cipher based on the knowledge of number theory, including the use of Euler's theorem. Therefore, knowing the theorem well and using it in specific mathematical applications is important. The aim of our paper is to show the validity of Euler's theorem by means of linear congruences and to present several specific tasks which are suitable to be solved using Euler's or Fermat's Little theorems and on which the principle of these theorems can be learned. Some tasks combine various knowledge from the field of number theory, and are specific by the fact that the inclusion of Euler's or Fermat's Little theorems to solve the task is not immediately apparent from their assignment.

References

Bose R. (2008). Information Theory, Coding and Cryptography. New Delhi, India: McGraw-Hill Publishing Company Limited, ISBN: 9780070669017.

Crilly T. (2007). 50 Mathematical Ideas You Really Need to Know. London: Quercus Publishing Plc, ISBN: 9781847240088.

Čižmár J. (2017). Dejiny matematiky – od najstarších čias po súčasnosť. Bratislava, Slovak Republic: PERFECT, ISBN: 9788080468293.

Ďuriš V. et al. (2019). Fibonacci Numbers and Selected Practical Applications in the Matlab Computing Environment. In: Acta Mathematica Nitriensia, ISSN 2453-6083, Vol. 5, No. 1, p. 14-22, DOI 10.17846/AMN.2019.5.1.14-22.

Koshy T. (2001). Elementary Number Theory with Applications. USA: Academic Press, 1st ed., ISBN: 9780124211711.

Znám Š. (1975). Teória čísel. Bratislava, Slovak Republic: SPN.

Jones G. A., Jones J. M. (1998). Elementary Number Theory. London: Springer, London, ISBN: 9783540761976.

Riesel H. (1994). Prime numbers and computer methods for factorisation. 2nd ed., Progress in Mathematics 126, Birkhauser.

Pommersheim J. E., Marks T. K., Flapan E. L. (2010). Number theory. USA: Wiley, 753 p., ISBN 978-0-470-42413-1.

Davydov U. S., Znám Š. (1972). Teória čísel – základné pojmy a zbierka úloh. Bratislava, Slovak Republic: SPN.

Apfelbeck A. (1968). Kongruence. Prague, Czech Republic: Mladá fronta.

Clifford A. P. (2011). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. New York, NY: Sterling Publishing, ISBN: 9781402757969.

Jackson T. (2017). Mathematics: An Illustrated History of Numbers (Ponderables: 100 Breakthroughs that Changed History) Revised and Updated Edition. New York, NY: Shelter Harbor Press, ISBN: 9781627950954.

Downloads

Published

2019-12-30

Issue

Vol. 37 (2019)

Section

Volume 37-2019

License

This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. The journal is committed to real and immediate open access for academic work. All of the journal articles are free to access immediately from the date of publication. There are no charge for readers to download articles and reviews for their own scholarly use. Benefits of open access for authors, include:

  • Free access for all users worldwide
  • Increased visibility and readership
  • Rapid publication
  • No spatial constraints
  • Authors can share their research papers on their personal web-pages (i.e. ResearchGate, etc.)

Ratio Mathematica publishes open access articles under the terms of the Creative Commons Attribution (CC BY) License. The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed. Copyright on any research article published by Ratio Mathematica is retained by the author(s). Authors grant Ratio Mathematica a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified. 
For further information visit: https://creativecommons.org/licenses/by/3.0/. The following logo will appear in any paper that will be published from the volume number 31/2016 onwards.