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Prime divisors of Lucas sequences. (English) Zbl 0913.11048

Let \(d>1\) be a squarefree integer and \(\mathbb{K}= \mathbb{Q}(\sqrt{d})\) a real quadratic field. Let \(\varepsilon= a+b\sqrt{d}\) be a fundamental unit in the ring of integers of \(\mathbb{K}\), and \(\overline{\varepsilon}\) its conjugate. The Lucas sequence associated with \(\mathbb{K}\) is the integer sequence defined by \[ X_{\mathbb{K}}= \{\text{Tr}_{\mathbb{K}/\mathbb{Q}} (\varepsilon^n) \}_{n=0}^\infty= \{\varepsilon^n+ \overline{\varepsilon}^n \}_{n=0}^\infty. \] The authors show that the set of prime numbers \(p\) that divide some term of the sequence \(X_{\mathbb{K}}\) has a natural density \(\delta_{\mathbb{K}}\) and determine it for each \(\mathbb{K}\).
Reviewer: P.Kiss (Eger)

MSC:

11R45 Density theorems
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11R11 Quadratic extensions
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