elementarynumbertheory

Elementary number theory is the study of integers and integer-valued functions using accessible, or elementary, methods. It emphasizes basic concepts such as divisibility, prime numbers, greatest common divisor, and the structure of the integers under addition and multiplication. The subject seeks to understand how integers behave through problems that can be solved with straightforward theorems, proofs, and constructive algorithms, without requiring heavy machinery from analysis or algebraic geometry. It forms the foundation for more advanced areas in number theory.

Key topics include divisibility theory, the Euclidean algorithm for gcds, primality and factorization, the sieve of Eratosthenes, modular arithmetic and congruences, linear congruences and the Chinese remainder theorem, Fermat's little theorem and Euler's theorem, Wilson's theorem, and quadratic residues and reciprocity. Important families of problems are Diophantine equations such as Pythagorean triples, Pell's equation, and, more generally, additive and multiplicative problems about representing numbers. Techniques are often constructive: algorithms for computing gcds, modular inverses, and continued fractions.

In addition to pure inquiry, elementary number theory has connections to cryptography, coding theory, and computer science, illustrating the practical relevance of integer properties. Historically, the subject traces to ancient mathematics and was advanced by classic works of Euclid, Fermat, Euler, and Gauss, among others, with modern developments such as rapid primality tests and integer factorization methods continuing to be active areas of study.