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For an integer $m\ge 2$, the Fibonacci entry point $z(m)$ is the smallest positive index with $m\mid F_{z(m)}$. It records the first zero residue, whereas the Pisano period records the first return of the whole consecutive pair $(F_n,F_{n+1})$ to $(0,1)$.

For $m=3$, the first positive divisible term is $F_4=3$, so $z(3)=4$; the pair period is eight. Existence follows because the pair update is an invertible permutation of a finite set of residue pairs, so its orbit returns to $(0,1)$. By Fibonacci gcd reduction, $m\mid F_n$ holds precisely when $z(m)\mid n$.