“Mathematics is the queen of the sciences and number theory is the queen of mathematics.” — Gauss
Number Theory is a discipline of pure mathematics concerned with the properties of numbers especially integers and their problem. Number theory is a vast filed subdivided into several branches. Elementary number theory is also known as arithmetic or the higher arithmetic. In elementary number theory, we study the integers without use of complex mathematical techniques such as questions of divisibility, using Euclidean algorithm in computation of greatest common divisors, investigation of perfect numbers and congruence, integer factorizations into prime numbers, etc.
Number theory has its root in ancient Roman and Indian civilization. It was used by Greeks to finding integer solutions to linear indeterminate equation, while it was extensively studied by Indian scholars and Aryabhatt arranged its theorem in a systematic order in his book named Aryabhatiya. In modern time it was redeveloped by Legendre and Gauss and they also started its uses to solve the question regarding integers.
Number theory is divided in sub fields based on the questions and their solution methods. Its main divisions are Elementary number theory, Analytic number theory, Algebraic number theory, Geometry of numbers, Combinatorial number theory, Computational number theory, Arithmetic algebraic geometry, Arithmetic topology, Arithmetic dynamics, Applied number theory, Modular forms (also known as complex analytic functions), Prime number theory, Public-key cryptography, Residue number system.
Number theory covers various topics of every field and is a wide subject who has become an essential need of the science and is used broadly almost in every field of the science and technology as well as our daily life subject like economics, engineering, computer programming etc.