Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

Number theory is a vast area of mathematics with several different flavours. Core questions that are dealt with in are approximating the density of (be it in all or in some residue class modulo a prime) and studying the or more general L-functions. In algebraic number theory, behaviour of ideals under extensions and contractions is one of the central problems. The subject is the study of integer solutions to certain, mostly polynomial, equations such as Pell's equation. Transcendental number theory is one other challenging subject which starts from transcendence of $e$, $\pi$.

A very popular subject of study, the field of number theory is rich in open problems, such as the , the Ramanujan conjecture, the Goldbach conjecture, and the (non-)existence of odd perfect numbers.