Educational > Pure Maths

Theory of Numbers


Number theory is the theory of the positive integers. It is based on ideas such as divisibility and congruence. Its fundamental theorem states that each positive integer has a unique prime factorization. In some ways it is the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated (but is yet to be proved or disproved). In other ways it is the least accessible discipline; for example, Wiles' proof that Fermat's equation has no nontrivial solutions requires understanding automorphic forms, which though intrinsic to nature have not found a place in physics or the general public discourse.

Masuma Parvin:
Number theory is one of the oldest branches of pure mathematics, and one of the largest. Of course, it concerns questions about numbers, usually meaning whole numbers or rational numbers (fractions).

Elementary number theory involves divisibility among integers -- the division "algorithm", the Euclidean algorithm (and thus the existence of greatest common divisors), elementary properties of primes (the unique factorization theorem, the infinitude of primes), congruences (and the structure of the sets  Z/nZ  as commutative rings), including Fermat's little theorem and Euler's theorem extending it. But the term "elementary" is usually used in this setting only to mean that no advanced tools from other areas are used -- not that the results themselves are simple. Indeed, a course in "elementary" number theory usually includes classic and elegant results such as Quadratic Reciprocity; counting results using the Möbius Inversion Formula (and other multiplicative number-theoretic functions); and even the Prime Number Theorem, asserting the approximate density of primes among the integers, which has difficult but "elementary" proofs. Other topics in elementary number theory -- the solutions of sets of linear congruence equations (the Chinese Remainder Theorem), or solutions of single binary quadratic equations (Pell's equations and continued fractions), or the generation of Fibonacci numbers or Pythagorean triples -- turn out in retrospect to be harbingers of sophisticated tools and themes in other areas.

Number theory is a very important branch of pure mathematics devoted primarily to the study of the natural numbers and the integers. It is sometimes called "The Queen of Mathematics"


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