Injectivity and surjectivity of Collatz functions Original Research Article

The paper studies conditions for injectivity and surjectivity of generalized Collatz functions. Necessary and sufficient conditions for each property, involving the complete set of parameters, are derived. The injectivity condition is formulated as a property of gcd matrices and the surjectivity condition is shown to be related to covering congruences. Furthermore, it is proved that injectivity implies ∑i=0d−11mi⩽1 while surjectivity implies ∑i=0d−11mi⩾1. Next, some implications of injectivity and surjectivity are discussed. In particular, it is shown that any injective Collatz function which is not surjective has divergent trajectories.