# Teaching Congruences in Connection with Diophantine Equations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Cryptography and Congruences

## 3. Results

- reflex: $a\equiv a\left(\mathrm{mod}m\right)$
- symmetric: $a\equiv b\left(\mathrm{mod}m\right)\Rightarrow b\equiv a\left(\mathrm{mod}m\right)$
- transitive: $a\equiv b\left(\mathrm{mod}m\right)\wedge b\equiv c\left(\mathrm{mod}m\right)\Rightarrow a\equiv c\left(\mathrm{mod}m\right)$

## 4. Research Methodology

**Hypothesis**

**(H1):**

**Hypothesis**

**(H2):**

## 5. Research Results

#### 5.1. Analysis of the Success of Respondents in Solving Tasks

_{1}that the median Y is greater. We implemented the test in the STATISTICA program. After entering the input data, we got the following results into the output set of the computer: the value of the test criterion of the one-sample Wilcoxon test (Z = 5.373) and the value of the probability p (p = 0.000). We evaluated the test using the value of p (p is the probability of an error we make when we reject the tested hypothesis). If the calculated value of the probability p is sufficiently small (p < 0.05 or p < 0.01), we reject the tested hypothesis (at the significance level 0.05 or 0.01). As the calculated value of the probability p < 0.01, at the level of significance α = 0.01 we reject the tested hypothesis H

_{0}. This means that by taking over or by supplementing the curriculum focused on the relationship between Diophantine equations and congruences in the optional seed, the level of students’ knowledge in the field of congruences increased statistically significantly.

#### 5.2. Analysis of the Length of Time Required to Solve Tasks

_{0}. This means that by teaching the curriculum focused on the relationship between Diophantine equations and congruences in the selective seminar, the time that students needed in the post-test to solve problems was statistically significantly reduced.

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- English, L.D. Promoting interdisciplinarity through mathematical modelling. ZDM
**2009**, 41, 161–181. [Google Scholar] [CrossRef] - Prawat, R.S. Current self-regulation views of learning and motivation viewed through a Deweyan lens: The problems with dualism. Am. Educ. Res. J.
**1998**, 351, 99–224. [Google Scholar] [CrossRef] - Marton, F.; Watkins, D.; Tang, C. Discontinuities and continuities in the experience of learning: An interview study of high-school students in Hong Kong. Learn. Instr.
**1997**, 7, 21–48. [Google Scholar] [CrossRef] - Gainsburg, J. The mathematical modeling of structural engineers. Math. Think. Learn.
**2006**, 8, 3–36. [Google Scholar] [CrossRef] - Noss, R.; Hoyles, C.; Pozzi, S. Abstraction in expertise: A study of nurses’conceptions of concentration. J. Res. Math. Educ.33(3)
**2002**, 204–229. [Google Scholar] [CrossRef] - Zawojewski, J.S.; McCarthy, L. Numeracy in Practice. Princ. Leadersh.
**2007**, 7, 32–37. [Google Scholar] - Metlenkov, N. Dynamics of architectural education. Astra Salvensis
**2018**, 1, 657–667. [Google Scholar] - Modeste, S. Impact of informatics on mathematics and its teaching. In International Conference on the History and Philosophy of Computing; Springer: Cham, Switzerland, 2015; pp. 243–255. [Google Scholar]
- Hauser, U.; Komm, D. Interdisciplinary education in mathematics and informatics at Swiss high schools. Bull. EATCS
**2018**, 3, 67–78. [Google Scholar] - Zaykis, R.; Campbell, S.R. Number Theory in Mathematics Education: Perspectives and Prospects; Routledge: London, UK, 2011. [Google Scholar]
- Rittle-Johnson, B.; Schneider, M. Developing conceptual and procedural knowledge of mathematics. In Oxford Handbook of Numerical Cognition; Oxford University Press: Oxford, UK, 2015; pp. 1118–1134. [Google Scholar]
- Nixon, J. Methods for Understanding Turing Machine Computations. Math. Aeterna
**2013**, 3, 709–738. [Google Scholar] - Mushtaq, M.F.; Jamel, S.; Disina, A.H.; Pindar, Z.A.; Shakir, N.S.A.; Deris, M.M. A Survey on the Cryptographic Encryption Algorithms. Int. J. Adv. Comput. Sci. Appl.
**2017**, 8, 333–344. [Google Scholar] - Coutinho, S.C. The Mathematics of Ciphers: Number Theory and RSA Cryptography, 1st ed.; A. K. Peters: Natick, MA, USA, 1999; p. 198. ISBN 9781568810829. [Google Scholar]
- Kleinjung, T.; Aoki, K.; Franke, J.; Lenstra, A.K.; Thomé, E.; Bos, J.W.; Gaudry, P.; Kruppa, A.; Montgomery, P.L.; Osvik, D.A.; et al. Factorization of a 768-Bit RSA Modulus. In Advances in Cryptology—CRYPTO 2010. Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2010; Volume 6223. [Google Scholar] [CrossRef][Green Version]
- Garey, M.R.; Johnson, D.S. Computers and Intractability—A Guide to the Theory of NP-Completeness; W. H. Freeman & Co.: New York, NY, USA, 1990; ISBN 0716710455. [Google Scholar]
- Pommersheim, J.E.; Marks, T.K.; Flapan, E.L. Number Theory; Wiley: Hoboken, NY, USA, 2010; p. 753. ISBN 978-0-470-42413-1. [Google Scholar]
- Koshy, T. Elementary Number Theory with Applications, 1st ed.; Academic Press: Cambridge, MA, USA, 2001; ISBN 9780124211711. [Google Scholar]
- Kodl, J.; Trojan, V. E-signature. Signatures in the electronic environment of communication networks. Vesmír
**2000**, 79, 611–613. [Google Scholar] - Ďuriš, V. Notes on Number Theory, 1st ed.; Verbum: Prague, Czech Republic, 2020; p. 141. ISBN 978-80-87800-63-8. [Google Scholar]
- Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed.; Springer: New York, NY, USA, 2012; p. 482. ISBN 978-0-8176-8298-9. [Google Scholar] [CrossRef]
- Jones, G.A.; Jones, J.M. Elementary Number Theory; Springer: London, UK, 1998; ISBN 9783540761976. [Google Scholar]
- Znám, Š. Number Theory; SPN: Bratislava, Slovak Republic, 1975. [Google Scholar]
- Dahlin, B.; Watkins, D. The role of repetition in the processes of memorizing and understanding: A comparison of the views of German and Chinese secondary school students in Hong Kong. Br. J. Educ. Psychol.
**2000**, 70, 65–84. [Google Scholar] [CrossRef] - Li, S. Does practice make perfect? Learn. Math.
**1999**, 19, 33–35. [Google Scholar] - Canobi, K.H. Concept-procedure interactions in children’s addition and subtraction. J. Exp. Child Psychol.
**2009**, 102, 131–149. [Google Scholar] [CrossRef] - McNeil, N.M.; Fyfe, E.R.; Petersen, L.A.; Dunwiddie, A.E.; Brletic-Shipley, H. Benefits of practicing 4=2+2: Nontraditional problem formats facilitate children’s understanding of mathematical equivalence. Child Dev.
**2001**, 82, 1620–1633. [Google Scholar] [CrossRef] - McNeil, N.M.; Chesney, D.L.; Matthews, P.G.; Fyfe, E.R.; Petersen, L.A.; Dunwiddie, A.E.; Wheeler, M.C. It pays to be organized: Organizing arithmetic practice around equivalent values facilitates understanding of math equivalence. J. Educ. Psychol.
**2012**, 104, 1109. [Google Scholar] [CrossRef][Green Version] - McNeil, N.M.; Fyfe, E.R.; Dunwiddie, A.E. Arithmetic practice can be modified to promote understanding of mathematical equivalence. J. Educ. Psychol.
**2015**, 107, 423. [Google Scholar] [CrossRef] - Cuarto, P.M. Algebraic Algorithm for Solving Linear Congruences: Its Application to Cryptography. Asia Pac. J. Educ. Arts Sci.
**2014**, 1, 34–37. [Google Scholar] - Ďuriš, V. Solving Some Special Task for Arithmetic Functions and Perfect Numbers. In Proceedings of the 19th Conference on Applied Mathematics, Bratislava, Slovakia, 6–24 February 2020; pp. 374–383. [Google Scholar]
- Nunes, T. Ethnomathematics and everyday cognition. In Handbook of Research on Mathematics Teaching and Learning; Grouws, D.A., Ed.; Macmillan: New York, NY, USA, 1992; pp. 557–574. [Google Scholar]
- Kilpatrick, J.; Swafford, J.; Findell, B. Adding It Up: Helping Children Learn Mathematics; National Research Council, Ed.; National Academy Press: Washington, DC, USA, 2001; Volume 2101. [Google Scholar]
- Fan, L.; Bokhove, C. Rethinking the role of algorithms in school mathematics: A conceptual model with focus on cognitive development. ZDM
**2014**, 46, 481–492. [Google Scholar] [CrossRef] - Hiebert, J.; Carpenter, T.P. Learning and teaching with understanding. In Handbook of Research on Mathematics Teaching and Learning; Grouws, D.A., Ed.; Macmillan: New York, NY, USA, 1992; pp. 65–97. [Google Scholar]
- Smith, J.C. Connecting undergraduate number theory to high school algebra: A study of a course for prospective teachers. In Proceedings of the 2nd International Conference on the Teaching of Mathematics, Crete, Greece, 1–6 July 2002; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2002; pp. 1–8. [Google Scholar]
- Alibali, M.W. How children change their minds: Strategy change can be gradual or abrupt. Dev. Psychol.
**1999**, 35, 127–145. [Google Scholar] [CrossRef] - Lemaire, P.; Siegler, R.S. Four aspects of strategic change: Contributions to children’s learning of multiplication. J. Exp. Psychol. Gen.
**1995**, 124, 83–97. [Google Scholar] [CrossRef] [PubMed] - Popat, S.; Starkey, L. Learning to code or coding to learn? A systematic review. Comp. Educ.
**2019**, 128, 365–376. [Google Scholar] [CrossRef] - Rittle-Johnson, B.; Star, J.R.; Durkin, K. Developing procedural flexibility: Are novices prepared to learn from comparing procedures? Br. J. Educ. Psychol.
**2012**, 82, 436–455. [Google Scholar] [CrossRef] [PubMed] - Silver, E.A.; Alacaci, C.; Stylianou, D.A. Students’ performance on extended constructed-response tasks. In Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress; Silver, E.A., Kenney, P.A., Eds.; National Council of Teachers of Mathematics: Reston, VA, USA, 2000; pp. 301–341. [Google Scholar]
- Schneider, M.; Rittle-Johnson, B.; Star, J.R. Relations between conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Dev. Psychol.
**2011**, 47, 1525–1538. [Google Scholar] [CrossRef] [PubMed][Green Version]

Message from Sender | x | y Encrypted Message | Decrypted Message | Message at the Recipient |
---|---|---|---|---|

R | 18 | 46 | 18 | R |

E | 5 | 26 | 5 | E |

X | 24 | 52 | 24 | X |

Problem | Z | p-Value |
---|---|---|

1 | 2.950 | 0.003 * |

2 | 2.094 | 0.036 * |

3 | 4.372 | 0.000 * |

4 | 5.232 | 0.000 * |

Problem | Z | p-Value |
---|---|---|

1 | 4.022 | 0.000 * |

2 | 3.456 | 0.001 * |

3 | 2.187 | 0.029 * |

4 | 2.113 | 0.035 * |

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**MDPI and ACS Style**

Viliam, Ď.; Dalibor, G.; Anna, T.; Pavlovičová, G.
Teaching Congruences in Connection with Diophantine Equations. *Educ. Sci.* **2021**, *11*, 538.
https://doi.org/10.3390/educsci11090538

**AMA Style**

Viliam Ď, Dalibor G, Anna T, Pavlovičová G.
Teaching Congruences in Connection with Diophantine Equations. *Education Sciences*. 2021; 11(9):538.
https://doi.org/10.3390/educsci11090538

**Chicago/Turabian Style**

Viliam, Ďuriš, Gonda Dalibor, Tirpáková Anna, and Gabriela Pavlovičová.
2021. "Teaching Congruences in Connection with Diophantine Equations" *Education Sciences* 11, no. 9: 538.
https://doi.org/10.3390/educsci11090538