Pell and Lucas primes

J. V. Leyendekkers and A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 3, Pages 64–69
Full paper (PDF, 95 Kb)

Details

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 Pell and Lucas numbers, Pp and Lp with prime subscripts are compared in relation to the function (Kp ± 1) and for factors of the form (kp ± 1). It is found that digit sums give some guides to primality.

Keywords

• Pell numbers
• Lucas numbers
• Primality
• Digit sums

AMS Classification

• 11B39
• 11B50

References

1. Dubner, H., & Keller, W. (1999) New Fibonacci and Lucas Primes. Mathematics of Computation. 68, 417−427.
2. Forget, T. W., & Larkin, T. A. (1968). Pythagorean Triads of the form x, x + 1, z described by Recurrence Sequences. The Fibonacci Quarterly 6 (3): 94–104.
3. Horadam, A.F. (1971) Pell Identities. The Fibonacci Quarterly. 9 (3): 245–252, 263.
4. Leyendekkers, J. V., & Shannon, A. G. (1998) Fibonacci Numbers within Modular Rings. Notes on Number Theory and Discrete Mathematics. 4 (4): 165–174.
5. Leyendekkers, J. V., & Shannon, A. G. (2013) The Structure of the Fibonacci Numbers in the Modular Ring Z₅. Notes on Number Theory and Discrete Mathematics. 19(1), 66–72.
6. Leyendekkers, J. V., & Shannon, A. G. (2013) Fibonacci and Lucas Primes. Notes on Number Theory and Discrete Mathematics. 19(2): 49–59.
7. Leyendekkers, J. V., & Shannon, A. G. (2014) Fibonacci Primes. Notes on Number Theory and Discrete Mathematics. 20(2), 6–9.
8. Leyendekkers, J. V., & Shannon, A. G. (2014) Fibonacci Numbers with Prime Subscripts: Digital Sums for Primes versus Composites. Notes on Number Theory and Discrete Mathematics. 20(3), 45–49.
9. Leyendekkers, J. V., & Shannon, A. G. (2014) Fibonacci Number Sums as Prime Indicators. Notes on Number Theory and Discrete Mathematics, 20(4), 47–52.
10. Leyendekkers, J. V., Shannon, A. G. & Rybak, J.M. (2007) Pattern Recognition: Modular Rings and Integer Structure. North Sydney: Raffles KvB Monograph No.9.
11. McDaniel, W. (2002) On Fibonacci and Pell Numbers of the form kx²: Almost every term has a 4r + 1 prime factor. The Fibonacci Quarterly. 40(1), 41–42.
12. Ray, P. K. (2013) New Identities for the Common Factors of Balancing and Lucas-Balancing numbers. Int. J. of Pure and Applied Mathematics. 85(3), 487–494.
13. Ribenboim, P. (1999) Pell Numbers, Squares and Cubes. Publicationes Mathematicae-Debrecen. 54(1–2), 131–152.
14. Robbins, N. (1983) On Fibonacci Numbers of the Form PX² where P is Prime. The Fibonacci Quarterly. 21(3), 266–271.
15. Robbins, N. (1984) On Pell Numbers of the Form PX² where P is Prime. The Fibonacci Quarterly. 22(4), 340–348.
16. Watkins, J J. (2014) Number Theory: A Historical Approach. Princeton and Oxford: Princeton University Press, pp. 271–272.