J. V. Leyendekkers and A. G. Shannon

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

Volume 12, 2006, Number 3, Pages 1–9

**Full paper (PDF, 131 Kb)**

## Details

### Authors and affiliations

J. V. Leyendekkers

*The University of Sydney, 2006 Australia*

A. G. Shannon

*Warrane College, Kensington, NSW 1465,
& KvB Institute of Technology, North Sydney, NSW 2060, Australia *

### Abstract

An integer structure (IS) of the sum (*x*^{4} + *y*^{4}) is done using the modular ring Z_{6}. This sum generated many primes and the row structure of such primes is explored. The class functions of the composite factors of this sum are also given, and these, together with the associated row functions, illustrate why it is impossible to produce an integer to the fourth power from such sums. The overall results are consistent with those previously found with IS analysis.

### AMS Classification

- 11A41
- 11A07

### References

- Irving Adler, Nonegenarian Fibonacci Devotee, The Mathematical Intelligencer, 28 (1) (2006): 4.
- J.V. Leyendekkers, J.M. Rybak & A.G. Shannon, Integer Class Properties Associated with an Integer Matrix. Notes on Number Theory & Discrete Mathematics. 1 (2) (1995): 53-59.
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- J.V. Leyendekkers & A.G. Shannon, Analysis of Row Expansions within the Octic ‘Chess’ Modular Ring, Z
_{8}, Notes on Number Theory & Discrete Mathematics. 5 (3) (1999): 102-114. - J.V. Leyendekkers & A.G. Shannon, The Analysis of Twin Primes within Z
_{6}, Notes on Number Theory & Discrete Mathematics. 7 (4) (2001): 115-124 - J.V. Leyendekkers & A.G. Shannon, Powers as a Difference of Squares, Notes on Number Theory & Discrete Mathematics. 8 (3) (2002): 95-106.
- J.V. Leyendekkers & A.G. Shannon, The Row Structure of Squares in Modular Rings, Notes on Number Theory & Discrete Mathematics. 10 (1) (2004): 1-11
- J.V. Leyendekkers & A.G. Shannon, The Structure of Fibonacci Numbers in Modular Rings, Notes on Number Theory & Discrete Mathematics. 10 (1) (2004): 12-23.
- J.V. Leyendekkers & A.G. Shannon, Using Integer Structure to Calculate the Number of Primes in a Given Interval, Notes on Number Theory & Discrete Mathematics. 10 (3) (2004): 77-83.
- Hans Riesel. Prime Numbers and Computer Methods for Factorization. 2
^{nd}edition. Progress in Mathematics, Volume 126. Boston: Birkhäuser, 1994.

## Related papers

## Cite this paper

Leyendekkers, J. V., and Shannon, A. G. (2006). Integer structure analysis of primes and composites from sums of two fourth powers. *Notes on Number Theory and Discrete Mathematics*, 12(3), 1-9.