Plain-language explanation.
Number theory is the study of integers — the whole numbers — and their properties. Once considered pure mathematics with no practical use, number theory now underpins the cryptographic systems that secure all modern digital communication.
Core concepts and standard treatment.
Core number theory covers divisibility and the division algorithm, prime numbers (Fundamental Theorem of Arithmetic — unique prime factorisation), GCD and LCM (Euclidean algorithm — extended Euclidean algorithm), modular arithmetic (congruences, Fermat's Little Theorem, Euler's theorem, Euler's totient function φ(n)), the Chinese Remainder Theorem, and quadratic residues (Legendre symbol, quadratic reciprocity — Gauss's gem of arithmetic).
Deeper theory, debates and edge cases.
Advanced number theory covers analytic number theory (Riemann zeta function, prime counting function π(x), Prime Number Theorem — ψ(x) ~ x; Dirichlet's theorem on primes in arithmetic progressions), algebraic number theory (rings of integers, ideals, unique factorisation in Dedekind domains, class groups, quadratic fields), elliptic curves over finite fields (group law, Weierstrass form, Hasse's theorem, the discrete logarithm problem), and unsolved problems (Riemann Hypothesis, Goldbach Conjecture, Twin Prime Conjecture, Birch and Swinnerton-Dyer Conjecture).
How it is applied in practice.
At the cryptographer and security engineer level, number theory is directly applied: RSA encryption (key generation via prime factorisation — trial division, Miller-Rabin primality testing; key exchange — Diffie-Hellman); elliptic curve cryptography (ECDSA, ECDH — Bitcoin uses secp256k1; TLS 1.3 uses P-256, X25519); lattice-based post-quantum cryptography (Learning With Errors — LWE; NIST PQC standardisation — CRYSTALS-Kyber, Dilithium); and blockchain (SHA-256 hash function, Merkle trees, zero-knowledge proofs — zk-SNARKs using elliptic curve pairings).