Number theory is the study of numbers, primarily positive integers, sometimes negative integers and occasionally algebraic integers. With cryptography and cryptanalysis becoming very important in our modern lives, number theory, especially dealing with the factorization of large prime numbers, has become a subject of great practical interest.
Elementary number theory uses only "elementary" methods from arithmetic, geometry and algebra; however, proofs in elementary number theory can be quite complicated. Analytic number theory uses real and complex analysis. Combinatorial number theory uses number theoretic methods to tackle problems in combinatorics. Computational number theory concerns itself with developing fast, efficient algorithms to tackle number theory problems that are far too computationally intensive for humans (such as the search for odd perfect numbers).