SUMMARY
The discussion focuses on computing the least residue of \(3^{215} \mod 65537\), where 65537 is identified as a Fermat's prime. Participants reference Euler's theorem, Fermat's little theorem, and Wilson's theorem in their attempts to solve the problem. A key clarification is made regarding the exponent, where \(3^{215}\) is corrected to \(3^{2^{15}}\), leading to the conclusion that \(3^{2^{15}} \equiv -1 \mod 65537\).
PREREQUISITES
- Understanding of Fermat's primes, specifically the form \(F_n = 2^{2^n} + 1\)
- Familiarity with modular arithmetic and least residues
- Knowledge of Euler's theorem and Fermat's little theorem
- Basic algebraic manipulation of exponents in modular contexts
NEXT STEPS
- Study the properties of Fermat's primes and their applications in number theory
- Learn about modular exponentiation techniques for efficient computation
- Explore advanced topics in number theory, such as the Chinese Remainder Theorem
- Investigate the implications of Fermat's little theorem in cryptographic algorithms
USEFUL FOR
Mathematicians, computer scientists, and students interested in number theory, particularly those working with modular arithmetic and prime number properties.