The modular ring Z₅

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 2, Pages 28–33
Full paper (PDF, 46 Kb)

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Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney
NSW 2006, Australia

A. G. Shannon
Faculty of Engineering & IT, University of Technology Sydney
NSW 2007, Australia

Abstract

The characteristics of the modular ring Z₅ are discussed. Each of the five classes is specific for two right-end-digits (REDs), even and odd. This facilitates analysis of the primitive Pythagorean triples, namely, the factors of components and the structure that prevents the two minor components from having REDs of (1,4), (5,6), (5,0) or (9,0). The RED feature is also useful in solving quadratic equations and the quick identification of modular classes. The distribution of powers within Z₅ is complex compared with other modular rings. Even powers are restricted to three Classes for n = 4r₂ + 2 ( ̅2₄ ⊂ Z₄) but only two Classes when n = 4r₀ ( ̅0₄ ⊂ Z₄). This power distribution is also useful in the analysis of power triples.

Keywords

• Integer structure analysis
• Modular rings
• Triangular numbers
• Tetrahedral numbers
• Pentagonal numbers
• Pyramidal numbers

AMS Classification

• 11A07

References

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3. Leyendekkers, J.V., A.G. Shannon. Why are some right-end digits absent in primitive Pythagorean triples? Notes on Number Theory & Discrete Mathematics. Vol. 17, 2011, No. 1, 37-44.
4. Leyendekkers, J.V., A.G. Shannon. Why 3 and 5 are always factors of primitive Pythagorean triples. International Journal of Mathematical Education in Science and Technology. Vol. 42, 2011, 102-105.
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7. Leyendekkers, J.V., A.G. Shannon. The Structure of Even Powers in Z3: Critical Structure Properties that Prevent the Formation of Even-powered Triples Greater than Squares. In preparation.
8. Shannon, A.G., J.V. Leyendekkers. 2011. Fibonacci Pythagorean Patterns. International Journal of Mathematical Education in Science and Technology. In press.