SUMMARY
The equation x² + 1 ≡ 0 (mod 5³) has specific solutions derived from its reduction to x² + 1 ≡ 0 (mod 5). The roots of the latter are x = 2, 3, 7, 8, 12, etc., but not all are valid for the original equation. Valid solutions for 0 ≤ x < 125 are of the form y + 25n, where y = 7 or 18, and n ranges from 0 to 4. The principal roots identified are 57 and 68, as they satisfy the necessary conditions.
PREREQUISITES
- Understanding of modular arithmetic, specifically modulo operations.
- Familiarity with quadratic equations and their solutions in modular contexts.
- Knowledge of the Chinese Remainder Theorem for solving systems of congruences.
- Basic calculus concepts, particularly derivatives in relation to roots.
NEXT STEPS
- Study the Chinese Remainder Theorem for solving modular equations.
- Explore quadratic residues and non-residues in modular arithmetic.
- Learn about the implications of derivatives in modular equations.
- Investigate higher powers of primes in modular equations, particularly mod 5ⁿ.
USEFUL FOR
Mathematicians, students studying number theory, and anyone interested in solving modular equations and understanding their properties.