Integer structure analysis of odd powered triples: The significance of triangular versus pentagonal numbers

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 16, 2010, Number 4, Pages 6–13
Full paper (PDF, 168 Kb)

Details

Authors and affiliations

J. V. Leyendekkers
The University of Sydney, 2006, Australia

A. G. Shannon
Warrane College, The University of New South Wales
Kensington, NSW 1465, Australia

Abstract

Structural constraints prevent the difference of two odd cubes ever equaling an even cube. This is illustrated from the row structure of the modular ring Z₄. The critical structure factor is that the rows of integers, N², with 3 | N², follow the triangular numbers, whereas 3 ∤ N² rows follow the pentagonal numbers. This structural characteristic is the reason for the importance of primitive Pythagorean triples (in which either the smallest odd component or the even component always has a factor 3).

Keywords

• Primitive Pythagorean triples
• Triangular numbers
• Pentagonal numbers
• Modular rings
• Cubic triples

AMS Classification

• 11A41
• 11A07
• 11B39
• 11C99

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