A Fibonacci integral lattice approach to Pythagoras’ Theorem

Anthony G. Shannon and John N. Crothers
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 1, Pages 88–90
Full paper (PDF, 55 Kb)

Details

Authors and affiliations

Anthony G. Shannon
Warrane College, The University of New South Wales
Kensington, 2033, Australia

John N. Crothers
Warrane College, The University of New South Wales
Kensington, 2033, Australia

Abstract

Square integral lattices with basis vector pairs {(a, b), (–b, a)}, where a and b are successive Fibonacci numbers, are employed to develop intermediate convergence forms of Pythagoras’ Theorem for triangles with integral sides.

Keywords

• Integral lattices
• Basis vectors
• Fibonacci numbers
• Pythagorean triples

AMS Classification

• 11B39
• 11D09
• 11H06

References

1. Crothers, J. N. (2013) An introduction to simple modular lattices. Advanced Studies in Contemporary Mathematics. 23(4): 637–653.
2. Shannon, A.G., Horadam, A.F. (1971) A Generalized Pythagorean Theorem. The Fibonacci Quarterly, 9(3): 307–312.
3. Shannon, A.G., Horadam, A.F. (1994) Arrowhead curves in a tree of Puythagorean triples.  Int. J. of Mathematical Education in Science and Technology. 25(2): 255–261.
4. Atanassov, K., Atanassova, V., Shannon, A. & Turner, J. (2002) New Visual Perspectives on Fibonacci Numbers. New Jersey, World Scientific.
5. Hunter, J. (1964) Number Theory. Edinburgh: Oliver and Boyd.
6. Horadam, A. F. (1961) A Generalized Fibonacci Sequence. American Mathematical Monthly, 58(5): 455-459.
7. Dickson, L. E. (1952) History of the Theory of Numbers, Volume 1. New York: Chelsea.
8. Shannon, A. G., & Horadam, A.F. (1973) Generalized Fibonacci Number Triples, 80(2): 187–190.