Number Theory Fundamentals

Explore linear congruences and modular arithmetic, learning when division is valid via gcd, and how to solve equations like r x + s ≡ t (mod m) with examples.

Learn about quadratic residues, determine when numbers have square roots modulo m, and use residue patterns to assess solvability of quadratic congruences and derive contradictions.

Learn divisor theory from prime factorization to counting and summing divisors, and explore perfect, abundant, and deficient numbers. Apply Fermat's theorem to period of a modulo p and primitive roots.

Explore Euler's totient function and its use in generalizing Fermat's theorem, derive phi(p^k) and multiplicativity, and learn Wilson's theorem with prime moduli.

Explore Dirichlet's theorem: infinitely many primes congruent to a mod m when gcd(a, m)=1, illustrated with the three mod four case and related concepts like v_p(n!).

Master powers modulo m with the successive squares method. Use Euler's theorem and gcd-based primality tests, including Carmichael numbers, to assess primality and perform efficient modular exponentiation.

Explore the law of quadratic reciprocity, Euler's criterion, and the Legendre symbol to decide quadratic residues and non-residues modulo odd primes, with practical examples.

Learn the sum of two squares theorem for primes, proving p is 2 or p ≡ 1 mod 4, and apply a divide-conquer method to general numbers.

Explore floor, ceiling, and fractional part functions, their basic properties, and function identities in number theory, including Hermite's identity and olympiad-style problem solving.