## 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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