On good initial values for the Lucas-Lehmer sequence (Q710501)

A sequence \((u_n)_{n\geq 0}\) of integers such that \(u_{n+1}=u_n^2-2\), \(n\geq 0\) is called Lucas-Lehmer. A good initial value \(u_0\) for such a sequence satisfies by definition \[ \forall p\geq p_0,\;p\,\text{prime}:\quad 2^p-1\,\text{is prime}\Leftrightarrow 2^p-1\mid u_{p-2}. \] The author proves that there are infinitely many good initial values for a Lucas-Lehmer sequence.