talteori

Talteori, or number theory, is a branch of pure mathematics dedicated to the study of integers and integer-valued functions. It concerns questions about divisibility, prime numbers, congruences, and the solutions of equations in integers, as well as the behavior of arithmetic functions defined on the integers. While its origins lie in antiquity, talteori now comprises a wide range of techniques and disciplines drawn from algebra, analysis, and geometry.

Key subfields include analytic number theory, which uses analytic methods to understand the distribution of primes and the behavior of L-functions; algebraic number theory, which studies numbers through their properties in algebraic extensions of the rationals; additive number theory, focusing on representations of numbers as sums; and computational/algorithmic number theory, which develops practical algorithms for primality testing, factorization, and cryptographic applications.

Historically, talteori dates to ancient Greece with Euclid’s proof that there are infinitely many primes. In the 18th and 19th centuries, Gauss and others developed deeper theory of congruences and prime distributions. The 20th century brought Dirichlet’s theorem on primes in arithmetic progressions and the development of analytic methods around the Riemann zeta function. In the late 20th and 21st centuries, proofs such as Wiles’s solution of Fermat’s last theorem and advances in modern algebraic geometry and automorphic forms have shaped the field.

Applications include modern cryptography (RSA, elliptic curve cryptography) and coding theory, which rely on number theoretic assumptions and algorithms. Beyond practical uses, talteori remains a central area of mathematical research, with many important conjectures—such as the Riemann Hypothesis and various Diophantine problems—guiding ongoing study.