Fibonacci Numbers Modulo 24 and the Heegner Discriminant Classes: An Equidistribution Theorem and a Prime-Index Density

We determine the exact behaviour of the Fibonacci sequence F (n) modulo 24. For odd indices n, the residues F (n) mod 24 lie in 1, 2, 5, 10, 13, 17, each with natural density exactly 1/6; in particular 5, 13, 17 occur with density exactly 1/2 (no bias within the admissible set). The set 5, 13, 17 is precisely the residues mod 24 of the six odd class-number-one discriminants (−7, −11, −19, −43, −67, −163). Over prime indices the density of F (p) ∈ 5, 13, 17 mod 24 is exactly 3/4, by the Pisano period and Dirichlet's theorem. All proofs are elementary; no modular-form mechanism and no claim regarding the Riemann Hypothesis. This version supersedes v1 (the earlier "≈58%" and "all nine" figures are withdrawn; correct values: 1/2 and six of nine).