Limiting density of the Fibonacci sequence modulo powers of a prime

2022/02/01 by Nicholas Bragman, Bragman, Nicholas, Eric Rowland +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2202.00704

Abstract

For a given prime p, we determine the limit, as λ→ ∞, of the density of residues modulo p^λ 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.

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