Prime Lehmer and Lucas numbers with composite indices (Q2868797)

Let \(U(P,Q)\) denote the Lucas sequence of parameters \(P\) and \(Q\), that is, \(U(P,Q)=\{U_n(P,Q)\}_{n\geq 0}\), where NEWLINE\[NEWLINE U_n(P,Q)=\frac{\alpha^n-\beta^n}{\alpha-\beta} NEWLINE\]NEWLINE where \(\alpha,~\beta\) are the roots of the quadratic \(x^2-PX+Q=0\). It is assumed that \(Q\neq 0\) and that \(\alpha/\beta\) is not a root of \(1\). Since \(U_n(P,Q)\mid U_m(P,Q)\), whenever \(n\mid m\), it follows that there should not be many composite indices \(n\) such that \(|U_n(P,Q)|\) is prime. Indeed, the authors' main result is that if \(n\) is composite and \(|U_n(P,Q)|\) is prime, then \(n\in \{4,6,8,9,10,15,25,26,65\}\). Furthermore, if \(n\in \{6,8,10,15,25,26,65\}\), then there are only finitely many pairs \((P,Q)\) such that \(|U_n(P,Q)|\) is prime and the authors determine them all. It is easy to find \((P,Q)\) such that \(|U_4(P,Q)|\) is prime and the authors conjecture that the same holds for \(|U_9(P,Q)|\) and show that this follows from Schinzel's Hypothesis H. Similar theorems are proved for the composite indices \(n\) such that the \(n\)th term of a Lehmer sequence of parameters \(L\) and \(M\) is prime. The proofs use the celebrated result of Bilu-Hanrot-Voutier on primitive prime divisors of members of Lucas and Lehmer sequences.