Decomposable polynomials in second order linear recurrence sequences

Primitive Prime Factors in Second-order Linear Recurrence Sequences

For a class of Lucas sequences {xn}, we show that if n is a positive integer then xn has a primitive prime factor which divides xn to an odd power, except perhaps when n = 1, 2, 3 or 6. This has several desirable consequences.

On Second-order Linear Recurrence Sequences: Wall and Wyler Revisited

Sequences of integers satisfying linear recurrence relations have been studied extensively since the time of Lucas [5], notable contributions being made by Carmichael [2], Lehmer [4], Ward [11], and more recently by many others. In this paper we obtain a unified theory of the structure of recurrence sequences by examining the ratios of recurrence sequences that satisfy the same recurrence relat...

Pseudoprimes for Higher-order Linear Recurrence Sequences

With the advent of high-speed computing, there is a rekindled interest in the problem of determining when a given whole number N > 1 is prime or composite. While complex algorithms have been developed to settle this for 200-digit numbers in a matter of minutes with a supercomputer, there is a need for simpler, more practical algorithms for dealing with numbers of a more modest size. Such practi...

On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or po...

Intersections of Second - Order Linear Recursive Sequences