Number theory is divided into several areas. As with any area of mathematics, the different sub-specialties are too numerous to list, but some of the largest areas include:
Analytic number theory employs techniques from real and complex analysis to answer questions about the natural numbers. The Prime Number Theorem and the related Riemann Hypothesis are part of analytic number theory, as are the proofs of the transcendence of numbers such as e or . The Riemann Hypothesis, proposed by the German mathematician Bernhard Riemann in 1859, remains one of the most famous unsolved problems in mathematics.
Algebraic number theory applies the methods of abstract algebra, group theory, field theory, and Galois theory to study the algebraic numbers—numbers that are the roots of polynomials with rational coefficients. Numbers that are not algebraic are transcendental—e and for example. Many concepts in algebraic number theory are studies modulo-p, where p is any prime number. This yields the p-adic numbers.
Combinatorial number theory applies combinatorial ideas such as partitions, covering systems, arithmetic progressions, and zero-sum problems, to number theoretic problems.
Computational number theory deals with algorithms relevant to number theory, such as integer factorization and primality testing algorithms. Computational number theory is intimately connected with cryptography.
At the heart of all the different branches of number theory is the Elementary number theory of Euclid, which studies the properties of the natural numbers without the use of techniques from other branches of mathematics. Questions in elementary number theory involve divisibility (greatest common divisors, for example), factorizations into prime numbers, special classes of natural numbers such as perfect numbers, multiplicative functions, and investigations of natural number sequences such as the Fibonacci numbers. Many famous conjectures in mathematics were stated in terms of elementary number theory, such as Goldbach’s conjecture, the Collatz conjecture, and Fermat’s Last Theorem. Those conjectures that have been proved have required techniques from outside elementary number theory.
Many questions in elementary number theory, and many of the applications of number theory, revolve around the notion of divisibility. If a, b and c are integers such that a = b *c, then b and c are said to divide (or are factors) of a, while a is said to be a multiple of b (as well as of c). The pipe symbol “|” denotes “divides” so we write b|a c|a. The divisors we are interested in most often for a given number are the prime divisors.
Remainders in division are just as important in number theory as divisors, in the form of modular arithmetic. Let a be an integer, and d be a positive integer. There are unique integers q, r with r {0,1,2,…,d-1} satisfying a = dq + r. The quantity r is called the remainder. We can calculate the remainder via the mod function. a mod b is a number between 0 and b –1 inclusive, which is the remainder of ab. A related notion is the modular congruence, which relates two numbers a, a’ to each other relative some base b. The notation a a’ (mod b) means that a and a’ have the same remainder when dividing by b.
The notion of congruence is useful for manipulating expressions involving the mod function. It lets us view modular arithmetic relative to a fixed base, allowing us to create a number system with a finite number of elements inside of which all the calculations can be carried out. The study of these finite number systems is called finite field arithmetic, and they are essential to most symmetric encryption algorithms.
Modular arithmetic is an essential aspect of number theory’s best known application, cryptography. For example, modular arithmetic's properties are at the heart of the RSA public key encryption algorithm.
No comments:
Post a Comment