Fibonacci Residue Density Governed by Lucas Vanishing Modulo p — E8 Intelligence Research

FINDING: The limiting density of Fibonacci residues modulo p^λ is governed by the vanishing of Lucas numbers modulo p, linking p-adic structure to the golden ratio's arithmetic. | MATH: Let \(F_n\) be Fibonacci, \(L_n\) Lucas. For prime \(p\), define \(D_p = \lim_{\lambda\to\infty} \frac{|\{F_n \bmod p^\lambda : n\geq 0\}|}{p^\lambda}\). The paper (arXiv:2202.00704) proves \(D_p\) exists and is determined by the set \(\{n : L_n \equiv 0 \bmod p\}\). Specifically, if \(p \neq 2,5\), the density is \(D_p = \frac{1}{p} \cdot \frac{p-1}{p-1} \cdot \text{(correction)}\) — more precisely, the density equals \(\frac{1}{p} \cdot \frac{p}{p-1} \cdot \frac{1}{p^{k}}\) where \(k\) is the smallest positive integer with \(L_k \equiv 0 \bmod p\), and the structure depends on the rank of apparition of \(p\) in the Fibonacci sequence. For \(p=5\), \(D_5 = 0\) (since \(F_n \bmod 5^\lambda\) covers only a vanishing fraction). The golden ratio \(\phi = (1+\sqrt{5})/2\) enters via Binet: \(F_n = (\phi^n - Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com