Number theory

Spirale Ulam 150

Plimpton 322

Diophantus-cover

Hevelius Selenographia frontispiece

Pierre de Fermat

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. Stemming from the simple operations of addition, subtraction, multiplication, and division, number theory dives into more complex realms such as the distribution of prime numbers, the solution of Diophantine equations, and the exploration of modular arithmetic. The field is often referred to as "higher arithmetic" due to its deep connections with fundamental arithmetic concepts, yet it encompasses a wide range of topics from elementary problems to sophisticated theories.

History

The origins of number theory trace back to ancient civilizations, including the Babylonians, Egyptians, and Greeks. Early work was primarily concerned with the properties of integers, divisibility, and the solutions to simple equations. The Greeks, particularly through the work of Euclid, made significant contributions with the systematic study of prime numbers and the introduction of Euclidean algorithm for finding the greatest common divisor.

During the 17th and 18th centuries, mathematicians such as Fermat, Euler, and Gauss expanded the scope of number theory. Fermat's Last Theorem, which he claimed to have proven but left no proof, and Euler's work on the distribution of primes laid the groundwork for future research. Gauss, often referred to as the "prince of mathematicians," made numerous contributions, including his work on quadratic reciprocity and the introduction of modular arithmetic.

In the 19th and 20th centuries, number theory saw substantial growth and formalization, with the development of algebraic number theory, analytic number theory, and the study of elliptic curves and modular forms. The advent of computers has also played a significant role in number theory, both in terms of computational number theory and in the proof of significant theorems, such as the resolution of Fermat's Last Theorem by Andrew Wiles in 1994.

Branches of Number Theory

Number theory is divided into several branches, each with its focus and methodologies.

Elementary Number Theory

Deals with the properties of integers under the usual operations of addition, subtraction, multiplication, and division. It includes the study of prime numbers, gcd and lcm, and the fundamental theorem of arithmetic.

Algebraic Number Theory

Explores the generalizations of integers to algebraic integers, studying number fields and their rings of integers. This branch is crucial for understanding the solutions to polynomial equations in integers and the structure of their solutions.

Analytic Number Theory

Uses tools from analysis to solve number-theoretical problems. It includes the study of the distribution of prime numbers, such as the Prime Number Theorem, and L-functions.

Computational Number Theory

Focuses on algorithms for solving number-theoretical problems, including primality testing, integer factorization, and the computation of elliptic curves. This branch has significant applications in cryptography.

Geometric Number Theory

Involves the study of numbers through geometric objects, such as lattices in Euclidean space and the geometry of numbers. It also includes the study of Diophantine geometry, which explores integer solutions to polynomial equations with several variables.

Important Concepts

• Prime Numbers: The building blocks of the integers, prime numbers are those greater than 1 that have no divisors other than 1 and themselves.
• Modular Arithmetic: A system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value, the modulus.
• Diophantine Equations: Polynomial equations with integer coefficients that seek integer solutions.
• Elliptic Curves: Algebraic curves used in number theory, cryptography, and other fields, which have a rich structure and deep implications for number theory.

Current Research

Research in number theory is vibrant and encompasses a wide range of topics, from the pursuit of understanding the distribution of prime numbers to the exploration of number-theoretical aspects in cryptography. The Langlands program, proposing deep connections between number theory and geometry, represents one of the most ambitious and far-reaching research agendas in modern mathematics.