The integer structure of the difference of two odd-powered odd integers

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 15, 2009, Number 3, Pages 14–20
Full paper (PDF, 163 Kb)

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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

The modular ring Z₄ was used to analyse the structure of the integer, N, obtained from xⁿ − yⁿ, x, y, n odd. The constraints on x and y associated with the probability of xⁿ − yⁿ = N = zⁿ (z even) were explored. When n ∈ ̅3₄ (n = 3, 7, 11, 15, …) the structure of N is 4r₀(4r₃ + 3) that is ̅0₄ × ̅3₄. When n ∈ ̅1₄ (n = 5, 9, 13, 17, …) the structure of N is 4r₀(4r₁ + 1) that is ̅0₄ × ̅1₄. The row structures and right-end-digit patterns of the rows of (x³ − y³) and z³ were compared and shown to be incompatible, as expected.

Keywords

• Primes
• Composites
• Modular rings
• Right-end digits
• Integer structure

AMS Classification

• 11A41
• 11A07

References

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