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Divisibility properties for Fibonacci and related numbers

Published online by Cambridge University Press:  23 January 2015

Jawad Sadek  and
Russell Euler
Jawad Sadek
Affiliation:
Dept. of Mathematics and Statistics, NWMSU, Maryville, MO, 64468 USA e-mail: reuler@nwmissouri.edu, jawads@nwmissouri.edu
Russell Euler
Affiliation:
Dept. of Mathematics and Statistics, NWMSU, Maryville, MO, 64468 USA e-mail: reuler@nwmissouri.edu, jawads@nwmissouri.edu
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Extract

Although it is an old one, the fascinating world of Fibonnaci numbers and Lucas numbers continues to provide rich areas of investigation for professional and amateur mathematicians. We revisit divisibility properties for t0hose numbers along with the closely related Pell numbers and Pell-Lucas numbers by providing a unified approach for our investigation.

For non-negative integers n, the recurrence relation defined by

with initial conditions

can be used to study the Pell (Pn), Fibonacci (Fn), Lucas (Ln), and Pell-Lucas (Qn) numbers in a unified way. In particular, if a = 0, b = 1 and c = 1, then (1) defines the Fibonacci numbers xn = Fn. If a = 2, b = 1 and c = 1, then xn = Ln. If a = 0, b = 1 and c = 2, then xn = Pn. If a =b = c = 2, then xn = Qn [1].

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 540 , November 2013 , pp. 461 - 464
Copyright
Copyright © The Mathematical Association 2013

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References

1. Horadam, A. F. and Mahon, Bro. J. M., Pell and Pell-Lucas polynomials, The Fibonacci Quarterly, 23.1 (February 1985) pp. 7–20.Google Scholar
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