SUMMARY
The discussion centers on finding the multiplicative order \( k \) for integers \( a \) and \( n \) where \( (a,n) = 1 \). It is established that \( a^{\phi(n)} \equiv 1 \mod n \) according to Euler's totient function \( \phi(n) \), but \( \phi(n) \) does not always yield the smallest \( k \). Participants agree that \( k \) must be a divisor of \( \phi(n) \) and highlight that there is no general formula for determining \( k \) except in specific cases such as \( a \equiv \pm 1 \mod n \). The discussion concludes that brute force remains the most effective method for finding \( k \) in general scenarios.
PREREQUISITES
- Understanding of Euler's totient function \( \phi(n) \)
- Knowledge of modular arithmetic
- Familiarity with primitive roots in number theory
- Basic concepts of divisors and their properties
NEXT STEPS
- Research the properties of Euler's totient function \( \phi(n) \)
- Learn about primitive roots and their significance in number theory
- Explore algorithms for brute force computation of multiplicative orders
- Study specific cases where \( k \) can be determined directly from \( a \) and \( n \)
USEFUL FOR
This discussion is beneficial for mathematicians, number theorists, and students studying advanced topics in number theory, particularly those interested in multiplicative orders and Euler's totient function.