### Topic Page: Number theory

**number theory**from

*The Penguin Dictionary of Mathematics*

The study of the arithmetic properties of integers and closely related number systems. Ancient Babylonian, Chinese, and Greek mathematicians were among the first to investigate numbers as interesting objects in themselves. Nowadays number theory is a large and many-sided discipline using, and stimulating the development of, sophisticated methods in several other areas of mathematics such as algebra and analysis. See Diophantine equation.

**number theory**

*The Columbia Encyclopedia*

branch of mathematics concerned with the properties of the integers (the numbers 0, 1, −1, 2, −2, 3, −3, …). An important area in number theory is the analysis of prime numbers. A prime number is an integer p>1 divisible only by 1 and p; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that have other divisors are called composite; examples are 4, 6, 8, 9, 10, 12, … . The fundamental theorem of arithmetic, the unique factorization theorem, asserts that any positive integer a is a product (a = p_{1} · p_{2} · p_{3} · · · p_{n}) of primes that are unique except for the order in which they are listed; e.g., the number 20 is the product 20 = 2 · 2 ·5, and it is unique (disregarding order) since 20 has this and only this product of primes. This theorem was known to the Greek mathematician Euclid, who proved that there are infinitely many primes. Analytic number theory has given a further refinement of Euclid's theorem by determining a function that measures how densely the primes are distributed among all integers. Twin primes are primes having a difference of 2, such as (3,5) and (11,13). The modern theory of numbers made its first great advances through the work of Leonhard Euler, C. F. Gauss, and Pierre de Fermat. It remains a major area of mathematical research, to which the most sophisticated mathematical tools have been applied.

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