Number theory studies the integers: their divisibility, primes, congruences, and patterns. It is one of the oldest branches of mathematics, dating back to Euclid and Diophantus. For centuries it was considered pure mathematics with no practical use. That changed with cryptography.

Prime numbers are central. The fundamental theorem of arithmetic states that every integer greater than 1 factors uniquely into primes. The distribution of primes remains a deep mystery, tied to the Riemann hypothesis, one of mathematics' great unsolved problems.

Modern number theory includes analytic number theory, algebraic number theory, and computational number theory. It underpins RSA encryption, error-correcting codes, and hashing algorithms. Once dismissed as useless, it now secures global communications. The subject rewards curiosity and rigor in equal measure.

• Study of integers and their properties
• Primes and factorization are central
• Riemann hypothesis is a major open problem
• Underpins modern cryptography