SUMMARY
The discussion focuses on proving that if \( (x, 7) = 1 \), then \( x^6 \equiv 1 \mod 7 \) using Fermat's Little Theorem. The theorem states that for a prime \( p \) and an integer \( a \) such that \( (a, p) = 1 \), it holds that \( a^{p-1} \equiv 1 \mod p \). In this case, applying the theorem with \( p = 7 \) confirms that \( x^6 \equiv 1 \mod 7 \). The user also explores the implications of this result for \( (x^3)^2 \equiv \pm 1 \mod 7.
PREREQUISITES
- Understanding of Fermat's Little Theorem
- Basic modular arithmetic
- Knowledge of prime numbers and their properties
- Concept of greatest common divisor (gcd)
NEXT STEPS
- Study the applications of Fermat's Little Theorem in number theory
- Explore proofs by induction in modular arithmetic
- Investigate the properties of congruences and their implications
- Learn about quadratic residues and their relevance in modular equations
USEFUL FOR
Students of number theory, mathematicians interested in modular arithmetic, and anyone studying proofs involving Fermat's Little Theorem.
