Modular Mirror: Efficient Residue Prediction in Primitive Root Cycles

Key Points

• To introduce a simplified method for calculating modular residues specifically in the latter half of a primitive root's power cycle.
• Developed the "Modular Mirror" rule for predicting residues
• Analyzed the residues in the context of odd prime numbers
• Compared the new method against traditional modular inverse techniques
• Found that the second half of residues are the additive inverses of the first half
• Demonstrated computational superiority of the new method over standard techniques
• Established that the calculation is efficient for primitive roots with order p - 1