SUMMARY
The discussion centers on demonstrating that \( a^{1728} \equiv 1 \mod p \) for primes \( p = 7, 13, \) and \( 19 \) using Fermat's Little Theorem. The theorem states that if \( p \) is a prime and \( p \nmid a \), then \( a^{p-1} \equiv 1 \mod p \). By calculating \( 1728 \mod 7, 1728 \mod 13, \) and \( 1728 \mod 19 \), the results are \( 6, 12, \) and \( 18 \) respectively, leading to the conclusion that \( a^{6} \equiv 1 \mod 7 \), \( a^{12} \equiv 1 \mod 13 \), and \( a^{18} \equiv 1 \mod 19 \), thereby confirming the theorem's application.
PREREQUISITES
- Understanding of Fermat's Little Theorem
- Basic modular arithmetic
- Knowledge of prime numbers
- Ability to compute modular reductions
NEXT STEPS
- Study the proofs of Fermat's Little Theorem
- Learn about modular exponentiation techniques
- Explore applications of Fermat's theorem in cryptography
- Investigate the properties of prime numbers in number theory
USEFUL FOR
Students studying number theory, mathematicians interested in modular arithmetic, and anyone looking to deepen their understanding of Fermat's Little Theorem.