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

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.


11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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