Modular Mirror: Efficient Residue Prediction in Primitive Root Cycles

This paper presents a simplified method for calculating modular residues in thesecond half of a primitive root’s power cycle. We demonstrate that for any oddprime p and primitive root a, the residues of the second half of the cycle (k > p−12 )are simply the additive inverses (reflections) of the first half. This ”Modular Mirror”rule allows for the determination of residues via a single subtraction (p−r), provingcomputationally superior to standard modular inverse methods when the order ofthe element is p − 1.