J. V. Leyendekkers and A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 17, 2011, Number 1, Pages 37—44

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## Details

### Authors and affiliations

J. V. Leyendekkers

Faculty of Science, The University of Sydney

Sydney, NSW 2006, Australia

A. G. Shannon

Faculty of Engineering & IT, University of Technology

Sydney, NSW 2007, Australia

### Abstract

Integer structure analysis in the Ring Z_{3} shows that the right-end digit (RED) couples (1,4), (5,6) and (5,0) for *x*^{2}, *y*^{2} in the primitive Pythagorean triple (pPt) in the equation *z*^{2} = *x*^{2} + *y*^{2} do not lead to the primitive form of triple. The rows of *x*^{2}, *y*^{2} with these REDs do not add to the required form for *z*^{2}. Since 3 ∤ *z* , the row of *z*^{2} must follow the pentagonal numbers. Common factors for *x*, *y* are also inconsistent with pPt formation so that the (*x*^{2}, *y*^{2}). RED (5,0) may be discarded directly.

### Keywords

- Integer structure analysis
- Modular rings
- Right-end-digits
- Primitive Pythagorean triples
- Triangular numbers
- Pentagonal numbers

### AMS Classification

- 11A41
- 11A07

### References

- Dickson, L.E. 1952.
*History of the Theory of Numbers. Volume 1.*New York: Chelsea. - Hardy, G.H., E.M. Wright. 1965.
*An Introduction to the Theory of Numbers.*Oxford: Clarendon Press. - Horadam, A.F., A.G. Shannon. 1988. Asveld’s Polynomials
*p*_{j}(*n*). In A.N. Philippou, A.F. Horadam, G.E. Bergum (eds).*Applications of Fibonacci Numbers*. Dordrecht: Kluwer, pp.163-176. - Leyendekkers, J.V., A.G. Shannon, J.M. Rybak. 2007.
*Pattern Recognition: Modular Rings and Integer Structure*. North Sydney: Raffles KvB Monograph No 9. - Leyendekkers, J.V., A.G. Shannon. 2011. The Number of Primitive Pythagorean Triples in a Given Interval.
*Notes on Number Theory & Discrete Mathematics*In press. - Leyendekkers, J.V., A.G. Shannon. 2011. Why 3 and 5 are always Factors of Primitive Pythagorean Triples.
*International Journal of Mathematical Education in Science & Technology.*42 (1): 102-105. - Leyendekkers, J.V., A.G. Shannon. 2011. Modular Rings and the Integer 3. In preparation.

## Related papers

## Cite this paper

APALeyendekkers, J., & Shannon, A.(2011). Why are some right-end digits absent in primitive Pythagorean triples?, Notes on Number Theory and Discrete Mathematics, 17(1), 37-44.

ChicagoLeyendekkers, JV, and AG Shannon. “Why Are Some Right-end Digits Absent in Primitive Pythagorean Triples?” Notes on Number Theory and Discrete Mathematics 17, no. 1 (2011): 37-44.

MLALeyendekkers, JV, and AG Shannon. “Why Are Some Right-end Digits Absent in Primitive Pythagorean Triples?” Notes on Number Theory and Discrete Mathematics 17.1 (2011): 37-44. Print.