Pellian sequences and squares

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 4, Pages 7–10
Full paper (PDF, 33Kb

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

Elements of the Pell sequence satisfy a class of second order linear recurrence relations which interrelate a number of integer properties, such as elements of the rows of even and odd squares in the modular ring Z₄. Integer Structure Analysis of this yields multiple-square equations exemplified by primitive Pythagorean triples, the Hoppenot equation and the equation for a sphere centred at the origin. The structure breaks down for higher powered triples so that solutions are blocked. However, Euler’s extension of Fermat’s Last Theorem does not work as the structure does permit multiple power equations such as a⁵ + b⁵ + c⁵ + d⁵ = e⁵.

Keywords

• Modular rings
• Integer structure analysis
• Pellian sequences
• Pythagorean triples
• Triangular numbers
• Pentagonal numbers.

AMS Classification

• 11A41
• 11A07.

References

1. Green, S. L. Algebraic Solid Geometry. Cambridge: Cambridge University Press, 1957, p. 60.
2. Leyendekkers, J. V., A. G. Shannon, J. M. Rybak, Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No. 9, 2007.
3. Leyendekkers, J. V., A. G. Shannon, Integer Structure Analysis of the Product of Adjacent Integers and Euler’s Extension of Fermat’s Last Theorem. Advanced Studies in Contemporary Mathematics. Vol. 17, 2008, No. 2, 221–229.
4. Leyendekkers, J. V., A. G. Shannon. Odd Powered Triples and Pellian Sequences. Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 3, 8–12.
5. Leyendekkers, J. V., A. G. Shannon. Geometrical and Pellian Sequences. Submitted.
6. Leyendekkers, J. V., A. G. Shannon. On the Sums of Multiple Squares. Notes on Number Theory and Discrete Mathematics, Vol. 18, 2012, No. 1, 9–15.
7. Melham, R. S. Alternating Sums of Fourth Powers of Fibonacci and Lucas Numbers. The Fibonacci Quarterly. Vol. 23, 2000, No. 3, 254–259.
8. Terr, D. Some Interesting Families of Primitive Pythagorean Triples. The Fibonacci Quarterly. Vol. 50, 2012, No. 1, 68–81