On Some New Arithmetic Properties of the Generalized Lucas Sequences

Some arithmetic properties of the generalized Lucas sequences are studied, extending a number of recent results obtained for Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These properties are then applied to investigate certain notions of Fibonacci, Lucas, Pell, and Pell–Lucas pseudoprimality, for which we formulate some conjectures.

[1]  P. Catarino,et al.  SUMS, PRODUCTS AND IDENTITIES INVOLVING k-FIBONACCI AND k-LUCAS SEQUENCES , 2014 .

[2]  Thomas Koshy,et al.  Fibonacci and Lucas Numbers with Applications: Koshy/Fibonacci , 2001 .

[3]  P. Catarino,et al.  On Dual $$\varvec{k}$$k-Pell Quaternions and Octonions , 2017 .

[4]  D. H. Lehmer.  An Extended Theory of Lucas' Functions , 1930 .

[5]  T. Andreescu,et al.  Number Theory: Structures, Examples, and Problems , 2009 .

[6]  Ángel Plaza,et al.  On the Fibonacci k-numbers , 2007 .

[7]  H. Hosoya.  What Can Mathematical Chemistry Contribute to the Development of Mathematics , 2013 .

[8]  S. Paul.  ON THE INFINITUDE OF LUCAS PSEUDOPRIMES , 1994 .

[9]  Milan Janjic,et al.  On Linear Recurrence Equations Arising from Compositions of Positive Integers , 2015, J. Integer Seq..

[10]  Thomas Koshy,et al.  Fibonacci and Lucas Numbers With Applications , 2018 .

[11]  P. Catarino,et al.  Two-by-two matrices involving k-Fibonacci and k-Lucas sequences , 2014 .

[12]  Manuel Torrilhon,et al.  Heliostat field optimization: A new computationally efficient model and biomimetic layout , 2012 .

[13]  C. Pomerance,et al.  Prime Numbers: A Computational Perspective , 2002 .

[14]  F. Al-Thukair,et al.  On Fibonacci and Lucas sequences modulo a prime and primality testing , 2018 .

[15]  S. Falcón.  On the Lucas triangle and its relationship with the k-Lucas numbers , 2012 .

[16]  A. F. Horadam,et al.  ON LUCAS PSEUDOPRIMES WHICH ARE PRODUCTS OF S PRIMES , 2007 .

[17]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987, JACM.

[18]  J. F. Rabago.  On k-Fibonacci Numbers with Applications to Continued Fractions , 2016 .

[19]  Karl Dilcher,et al.  A search for Wieferich and Wilson primes , 1997, Math. Comput..

[20]  P. Catarino.  A note involving two-by-two matrices of the k-Pell and k-Pell-Lucas sequences , 2013 .

[21]  J. H. Jaroma.  Note on the Lucas-Lehmer Test , 2004, Irish Mathematical Society Bulletin.

[22]  A. Rotkiewicz.  LUCAS AND FROBENIUS PSEUDOPRIMES , 2003 .

[23]  J. Kiefer,et al.  Sequential minimax search for a maximum , 1953 .

[24]  S. Schuster,et al.  Use of Fibonacci numbers in lipidomics – Enumerating various classes of fatty acids , 2017, Scientific Reports.

[25]  Titu Andreescu,et al.  Quadratic Diophantine Equations , 2015 .

[26]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .