SUMMARY
The discussion centers on proving that for any prime number p greater than or equal to 5, the expression p^2 + 2 is always composite. The analysis utilizes modular arithmetic, specifically examining the remainders when p is divided by 6. The cases for p = 6q, 6q+1, 6q+2, 6q+4, and 6q+5 demonstrate that p^2 + 2 results in composite numbers, while the case for p = 6q+3 indicates that p cannot be prime as it is divisible by 3. Thus, the conclusion is that p^2 + 2 is composite for all primes p ≥ 5.
PREREQUISITES
- Understanding of modular arithmetic
- Familiarity with prime numbers and their properties
- Basic algebraic manipulation skills
- Knowledge of composite numbers
NEXT STEPS
- Study modular arithmetic in depth, focusing on applications in number theory
- Explore properties of prime numbers and their distributions
- Learn about proofs in number theory, particularly those involving composites and primes
- Investigate the implications of divisibility rules in algebraic expressions
USEFUL FOR
Mathematics students, educators, and anyone interested in number theory, particularly those studying properties of prime and composite numbers.