The Fibonacci numbers are not an additive uniqueness set for multiplicative functions

Abstract

Let $(F_n{)}_{n\ge 0}$ be the Fibonacci sequence. We show that a multiplicative function $f$ satisfying \[ f(F_n+F_m)=f(F_n)+f(F_m)\qquad(n,m\geq 1) \] need not be the identity function, even when $f$ takes positive integer values. This answers negatively a question posed by Spiro in 1992. The smallest example presented here is obtained from $F_{31}=557\cdot 2417$. We prove the required divisibility equivalence, formulate an abstract prime-signature construction, and give a practical criterion producing further examples. We also describe the AI-assisted search that led to the construction and provide a reproducible certificate checker.

7 pages, AI-assisted