Arithmetic and Trigonometric Properties of Some Classical Recurrent Sequences

Abstract

The classical second-order recurrent sequences discussed in Chapter 2 have many interesting properties. In Section 3.1 we present arithmetic properties of the Fibonacci and Lucas sequences first found in [23]. Section 3.2 studies properties of Pell and Pell–Lucas sequences. We also discuss about certain notions of Fibonacci, Lucas, Pell, or Pell–Lucas primality. These results have been extended to generalized Lucas and Pell–Lucas sequences in our paper (Andrica and Bagdasar, Mediterr. J. Math. 2021, to appear). Section 3.3 presents factorizations of the terms of classical sequences as products of trigonometric expressions. These are derived from the complex factorizations of the general polynomials Un and Vn ( 2.60), and some involve the resultant of polynomials.