The local Dold congruence for Fibonacci and Lucas sequences

We discuss local properties of the classical Fibonacci and Lucas sequences, characterising in both cases those primes~$p$ for which the~$p$-part satisfies the Dold congruences or is realizable. We calculate an exact `repair’ multiplier in each case and determine the relative prime densities of the Dold, realizable, almost Dold, and almost realizable prime sets. In that order, the four densities are~$(\frac{3}{4},\frac{1}{3},1,\frac{1}{3})$ for the Lucas sequence and~$(\frac{1}{2},\frac{1}{2},1,1)$ for the Fibonacci sequence.