For a given prime p, we determine the limit, as \(\lambda \rightarrow \infty \) , of the density of residues modulo \(p^\lambda \) attained by the Fibonacci sequence. In particular, we show that this limiting density is related to zeros in the sequence of Lucas numbers modulo p. The proof uses a piecewise interpolation of the Fibonacci sequence to the p-adic numbers and a characterization of Wall–Sun–Sun primes p in terms of the p-adic absolute value of a number related to the p-adic golden ratio.
