Squares in generalizations of Lucas sequences. (Les carrés dans des généralisations des suites de Lucas.) (French) Zbl 1107.11008

Samuel, Pierre

J. Théor. Nombres Bordx. 16, No. 3, 693-703 (2004).

Let \(P\) and \(Q\) be natural relatively prime and odd numbers, such that \(D=P^2-4Q>0\). Let \(U_0=1\), \(U_1=1\), \(U_{n+2}=PU_{n+1}-QU_n\), \(v_0=2\), \(v_1=P\), \(V_{n+2} =PV_{n+1}-QV_n\). In the paper there have been studied some properties of sequence \(\{x_n\}_{n\geq 0}\) of positive integers satisfying the recursion formula \(x_{n+2}= Px_{n+1}-Qx_n\) that is a generalization of sequences \(\{U_n\}_{n\geq 0}\) and \(\{V_n\}_{n\geq 0}\). For example, the basic properties of the prime divisors of \(\{V_n\}_{n\geq 0}\) for \(n=3.2^j\) are discussed.

Reviewer: Krassimir Atanassov (Sofia)

MSC:

11B39

Fibonacci and Lucas numbers and polynomials and generalizations

Keywords:

Fibonacci number; Lucas number

References:

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