Modular Exponent Tower Collapse over Prime Squares

This short note explores a surprising property of exponential towers in modular arithmetic. When the base is a square of a prime number, the tower “collapses” to a much smaller exponent modulo the base. The proof uses basic ideas from number theory, including Fermat’s Little Theorem and the Lifting The Exponent Lemma, and works for any number coprime to the prime. A simple but elegant result that uncovers hidden patterns in modular exponentiation.