## 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.

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations

### Keywords:

Fibonacci number; Lucas number
Full Text:

### References:

 [1] J.H.E. Cohn, Eight Diophantine equations. Proc. London Math. Soc. (3), 16 (1966), 153-166. · Zbl 0136.02806 [2] J.H.E. Cohn, Some quartic Diophantine equations. Pacific J. of Math. 26, 2 (1968), 233-243. · Zbl 0191.04902 [3] W. Mc Daniel, P. Ribenboim, The square terms in Lucas sequences. J. Number Theory, 58, 1 (1996), 104-123. · Zbl 0851.11011 [4] J.P. Serre, Cours d’Arithmétique. Presses Univ. de France, 1970. · Zbl 0225.12002
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