Integer structure analysis of primes and composites from sums of two fourth powers

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 13, 2007, Number 2, Pages 1—9
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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 (x4 + y4) is done using the modular ring Z6. 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

  1. Irving Adler, Nonegenarian Fibonacci Devotee, The Mathematical Intelligencer, 28 (1) (2006): 4.
  2. 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.
  3. J.V. Leyendekkers, J.M. Rybak & A.G. Shannon, Analysis of Diophantine Properties Using Modular Rings with Four and Six Classes. Notes on Number Theory & Discrete Mathematics. 3 (2) (1997): 61-74.
  4. J.V. Leyendekkers & A.G. Shannon, Analysis of Row Expansions within the Octic ‘Chess’ Modular Ring, Z8, Notes on Number Theory & Discrete Mathematics. 5 (3) (1999): 102-114.
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  6. J.V. Leyendekkers & A.G. Shannon, Powers as a Difference of Squares, Notes on Number Theory & Discrete Mathematics. 8 (3) (2002): 95-106.
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  8. 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.
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  10. Hans Riesel. Prime Numbers and Computer Methods for Factorization. 2nd edition. Progress in Mathematics, Volume 126. Boston: Birkhäuser, 1994.

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Cite this paper

APA

Leyendekkers, J., & Shannon, A. (2007). Integer structure analysis of primes and composites from sums of two fourth powers. Notes on Number Theory and Discrete Mathematics, 13(2), 1-9.

Chicago

Leyendekkers, JV, and AG Shannon. “Integer Structure Analysis of Primes and Composites from Sums of Two Fourth Powers.” Notes on Number Theory and Discrete Mathematics 13, no. 2 (2007): 1-9.

MLA

Leyendekkers, JV, and AG Shannon. “Integer Structure Analysis of Primes and Composites from Sums of Two Fourth Powers.” Notes on Number Theory and Discrete Mathematics 13.2 (2007): 1-9. Print.

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