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

Viliam Ďuriš


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.


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

Full Text:



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.



  • There are currently no refbacks.

Copyright (c) 2019 Viliam Ďuriš

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.