On the prime density of Lucas sequences (Q679108)

By the prime density of an integer sequences \((A_n)_{n \geq 0}\) we mean the natural density of the set of all prime numbers dividing \(A_n\) for some \(n \geq 0\). Let \(D>1\) be a square-free integer such that the real quadratic field \(\mathbb{Q} (\sqrt D)\) has a fundamental unit with norm \(-1\), and let \(\varepsilon\) be any unit of \(\mathbb{Q} (\sqrt D)\). Then the prime density of \((\varepsilon^n+ \overline \varepsilon^n)_{n \geq 0}\) is explicitly calculated. As an application, the author determines the prime density of the Lucas sequence \((L_n)_{n \geq 0}\), given by \(L_0= 2\), \(L_1= P\) and \(L_n= PL_{n-1} +L_{n-2}\) for an arbitrary nonzero integer \(P\). The proof depends on the calculation of the degrees of several explicitly given radical extensions of \(\mathbb{Q}(\sqrt D)\).