Fibonacci and Lucas primes

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 19, 2013, Number 2, Pages 49—59
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Authors and affiliations

J. V. Leyendekkers
Faculty of Science, The University of Sydney
NSW 2006, Australia

A. G. Shannon
Faculty of Engineering & IT, University of Technology Sydney
NSW 2007, Australia

Abstract

The structures of Fibonacci numbers, Fn, formed when n equals a prime, p, are analysed using the modular ring Z5, Pascal’s Triangle as well as various properties of the Fibonacci numbers to calculate “Pascal-Fibonacci” numbers to test primality by demonstrating the many structural differences between the cases when Fn is prime or composite.

Keywords

  • Fibonacci sequence
  • Golden Ratio
  • Modular rings
  • Pascal’s triangle
  • Binet formula

AMS Classification

  • 11B39
  • 11B50

References

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  2. Benjamin, A. T. The Joy of Mathematics. The Great Courses, Chantilly, VA, 2007.
  3. Deakin, M. A. B. Theano: the World’s First Female Mathematician? Int. Journal of Mathematical Education in Science & Technology. Vol. 44, 2013, No. 3, 350–364.
  4. Kim, D. S., T. Kim, H. Y. Lee. p-adic q-integral on Zp associated with Frobenius-type Eulerian Polynomials and Umbral Calculus. Advanced Studies in Contemporary Mathematics. Vol. 23, 2013, No. 2, 243–251.
  5. Knuth, D. E. Art of Computer Programming, Volume 4. Addison-Wesley, New York, 2005, p. 50.
  6. Leyendekkers, J. V., A. G. Shannon. The Structure of the Fibonacci Numbers in the Modular Ring Z5. (Submitted).
  7. Livio, M. The Golden Ratio. Golden Books, New York, 2002.
  8. Lucas, E. Théorie des Fonctions Numériques Simplement Périodiques. American Journal of Mathematics. Vol. 1, 1878, 184–240.

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Cite this paper

APA

Leyendekkers, J. V., & Shannon, A. (2013). Fibonacci and Lucas primes. Notes on Number Theory and Discrete Mathematics, 19(2), 49-59.

Chicago

Leyendekkers, JV, and AG Shannon. “Fibonacci and Lucas Primes.” Notes on Number Theory and Discrete Mathematics 19, no. 2 (2013): 49-59.

MLA

Leyendekkers, Tieling, and AG Shannon. “Fibonacci and Lucas Primes.” Notes on Number Theory and Discrete Mathematics 19.2 (2013): 49-59. Print.

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