Limiting density of the Fibonacci sequence modulo powers of a prime

TL;DR

This paper investigates the asymptotic density of Fibonacci sequence residues modulo powers of a prime, revealing connections to Lucas numbers and p-adic properties, and employs p-adic interpolation and prime characterization techniques.

Contribution

It provides a new limit formula for Fibonacci residues modulo prime powers and links this to Lucas numbers and Wall-Sun-Sun primes using p-adic analysis.

Findings

Limit of Fibonacci residue density is characterized for prime powers.

Connection established between Fibonacci residues and Lucas number zeros.

Characterization of Wall-Sun-Sun primes via p-adic properties.

Abstract

For a given prime $p$, we determine the limit, as $\lambda \to \infty$, 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.